
% Period functions for Maass wave forms and cohomology
% Don Zagier, John Lewis, Roelof Bruggeman
%
% August 26, 2013

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\title{Period Functions for Maass Wave Forms and Cohomology}

\author{R.\,Bruggeman}
\address{Mathematisch Instituut Universiteit Utrecht,
3508 TA Utrecht,
Nederland}
\email{r.w.bruggeman@uu.nl}

\author{J.\,Lewis}
\address{Massachusetts Institute of Technology, Cambridge,
MA 02139, USA}
\email{jlewis@math.mit.edu}

\author{D.\,Zagier}
\address{Max-Planck-Institut f\"ur Mathematik, 53111 Bonn,
Deutschland; and Coll\`ege de France, 75005 Paris, France}
\email{don.zagier@mpim-bonn.mpg.de}

\date{August 26, 2013}

\begin{abstract} We construct explicit isomorphisms between spaces of
Maass wave forms and cohomology groups for discrete cofinite groups
$\Gamma\subset\mathrm{PSL}_2({\mathbb{R}})$.

In the case that $\Gamma$ is the modular group
$\mathrm{PSL}_2({\mathbb{Z}})$ this gives a cohomological framework
for the results in \emph{Period functions for Maass wave forms.~I,}
of {\sc J.\,Lewis} and {\sc D.\,Zagier} in Ann.\ Math.\ {\bf 153}
(2001), 191--258, where a bijection was given between cuspidal Maass
forms and period functions.

We introduce the concepts of \emph{mixed parabolic cohomology group}
and \emph{semi-an\-a\-lytic vectors} in principal series
representation. This enables us to describe cohomology groups
isomorphic to spaces of Maass cusp forms, spaces spanned by residues
of Eisenstein series, and spaces of all $\Gamma$-invariant
eigenfunctions of the Laplace operator.

For spaces of Maass cusp forms we also describe isomorphisms to
parabolic cohomology groups with smooth coefficients and standard
cohomology groups with distribution coefficients. We use the latter
correspondence to relate the Petersson scalar product to the cup
product in cohomology.
\end{abstract}

\subjclass[2010]{ 11F37
%(Forms of half-integer weight; nonholomorphic
	 %modular forms)
11F67
%(Special values of automorphic $L$-series, periods of modular
	 %forms, cohomology, modular symbols)
 11F75 %(Cohomology of arithmetic groups)
22E40 %(Discrete subgroups of Lie groups)
 }

\keywords{ Maass form, period function, cohomology group, parabolic
cohomology, principal series, Petersson scalar product, cup product }



\maketitle


\tableofcontents



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\section*{Introduction} 
These notes proceed from the ideas and results
of \cite{LZ}, in which Maass forms for the full modular group were
studied, but now treating arbitrary Fuchsian groups and stressing the
cohomological interpretation. They can be read independently
of~\cite{LZ}.

The classical theory linking holomorphic automorphic forms to
cohomology starts with Eichler \cite{Ei57}, and Shimura
\cite{Sh59}. To an automorphic form $F$ on the upper half-plane  with 
even weight $k\geq 2$ one associates a cocycle with values in the
space of polynomial functions of degree at most $k-2$ by
$\ps_\gm(t) =
\int_{\gm^{-1}z_0}^{z_0} (t-\tau)^{k-2}\, F(\tau)\, d\tau$, with a
base point $z_0$ in the upper half-plane. If $F$ is a cusp form, one
can put the base point at~$\infty$. The coefficients of the resulting
polynomials are then values of the $L$-function of~$F$. All this has
important number theoretical consequences.
(See, e.g., Manin \cite{Ma73}.)

With the base point at $\infty$, the cocycle is, in the case of the
modular group $\SL_2(\ZZ)$, determined by its value on
$S=\rmatr0{-1}10$\,,
  \[ \ps_S (t) \= \int_0^{i\infty} (t-\tau)^{k-2}\, F(\tau)\,
  d\tau\,,\]
called the \emph{period function} (or \emph{period polynomial})
of~$F$, and the condition of being a cocycle is equivalent to the two
functional equations
  \be \label{AAA}
  \psi(t) + t^{k-2}\, \ps\bigl(-\frac1t\bigr) \= 0\,,\quad \psi(t) +
  t^{k-2}\, \ps\bigl(1-\frac1t\bigr) + (1-t)^{k-2}\,
  \ps\bigl(\frac1{1-t}\bigr) \= 0\,.\ee
In this case, it is also known that the map assigning to a cusp form
$F(\tau)$ the odd part of the polynomial $\ps_S$ is an isomorphism
between the space of cusp forms of weight~$k$ and the vector space of
odd polynomials $\ps(t)$ satisfying~\eqref{AAA}. An elementary
argument shows that this latter space can be characterized by a
single functional equation
  \[ \ps(t) \= \ps(t+1) \,+\, t^{k-2}\,\ps(1+1/t)\,.\]
The starting point of \cite{LZ} (see also the survey paper \cite{LZ1}
and \S2 of \cite{Za93})
is the observation that this functional equation is identical in form
to the relation
  \be\label{BBB} \ps( t ) \= \ps( t +1) \,+\, t^{-2s}\,\ps(1+1/ t) \ee
that occurred in the work of the second author \cite{Le97}, which gave
a bijection between the space of even Maass wave forms with spectral
parameter~$s$ on the full modular group and a class of holomorphic
functions satisfying~\eqref{BBB}. Since~\eqref{AAA} is just the
cocycle condition for $\SL_2(\ZZ)$, this immediately suggests the
possibility of describing Maass forms for arbitrary Fuchsian groups
 by an appropriate generalization of the functional
equation~\eqref{BBB} having an interpretation in terms of cohomology.


The principal goal of these notes is to carry out this
generalization by constructing explicit isomorphisms between, on the
one hand, spaces of Maass wave forms on discrete cofinite groups
$\Gm\subset G:=\PSL_2(\RR)$ and, on the other, certain cohomology
groups of~$\Gm$. 
Recall that a \indlbl{Mwfi}{wave form}\emph{Maass wave form}
(or simply \indlbl{Mfi}{Maass form}\emph{Maass form})
on $\Gm$ is a $\Gm$-invariant function on $\uhp$ satisfying
$\Dt u=\ld u$ for some $\ld\in \CC$, with polynomial growth. Here
$\uhp$ is the complex upper half-plane with the usual action of $G$
and $\Dt$ is the hyperbolic Laplace operator
$\Dt=-y^2(\p_x^2+\p_y^2)$. The Maass wave forms which are small at
the cusps (this is relevant only for $\Gm\backslash\uhp$ non-compact)
we call \indlbl{Mcfi}{Maass cusp form}\emph{Maass cusp
forms}.\footnote{In \cite{LZ} the word ``wave form`` was used for
cusp forms only.} The eigenvalue $\ld$ is most naturally written as
$s(1-s)$ for some $s\in\CC$ (\indlbl{spi}{spectral parameter}{\it
spectral parameter}), and our cohomological description of Maass wave
forms will depend on picking one of the two roots of this equation.
We assume throughout that $0<\re(s)<1$.

In~\cite{LZ} we studied the case of the full modular group
\indlbl{mgi0}{modular group, $\Gmod$}$\Gmod:=\PSL_2(\ZZ)$ in detail
and showed that the Maass cusp forms with eigenvalue $s(1-s)$ are
canonically in one-to-one correspondence with the real-analytic
functions $\,\psi\,:\,(0,\infty)\to\CC$ which satisfy
\be \ps(x)\=\ps(x+1)\,+\,(x+1)^{-2s}\, \ps\Bigl(\frac
x{x+1}\Bigr)\qquad(x>0)\label{3T}\ee
and for which both $\ps(x)$ and $x^{2s}\ps(x)$ are bounded. It turns
out that any such function can be written
(non-uniquely) as
\be \ps(x)\=h(x)\,-\,x^{-2s}\, h(-1/x)\qquad(x>0)\label{tunnel}\ee
for some real-analytic function $\,h\,:\,\RR\to\CC$ and that, when we
do this, the map
\be \rmatr0{-1}10\;\mapsto\;0\;,\qquad\rmatc1101\;\mapsto
    \;\bigl(x\mapsto\vphantom{|^2} h(x+1)-h(x)\bigr)\;,\label{ancoc}\ee
extends to a cocycle on $\Gmod$ with values in the analytic vectors
\indlbl{Vomi}{$\V\om s$}$\V\om s$ of a model of the principal series
representation $\V{}s$. Changing the choice of $h$ changes this
cocycle by a coboundary, and we get an isomorphism between the space
$\Mf^0_s(\Gmod)$ of Maass cusp forms on $\Gmod$ and a specific
subspace of $H^1(\Gmod;\V\om s)$. Our goal in these notes is to give
an analogous result for all~$\Gm$.

To achieve this, we will use several descriptions of the principal
series: The model indicated above consists of functions on the
boundary $\proj\RR$ of the upper half-plane. We shall also use models
of the principal series in functions on $\uhp$ itself. The relevant
investigations led to the paper~\cite{BLZ}. We shall recapitulate the
results we need in Sections \ref{ssect-prs} and~\ref{sect-bgtPt}.
Specifically, the well known \emph{Poisson transformation} realizes
the principal series representation with spectral parameter~$s$ as
the space $\E_s$ of all solutions on~$\uhp$ of the differential
equation $\Dt u = s(1-\nobreak s) u$. For the construction of the map
from cohomology to Maass forms, we introduce a \emph{transverse
Poisson transformation} $\tpoiss s$, which provides us with a model
of the principal series in a space of solutions of
$\Dt u = s(1-\nobreak s) u$ near the boundary of~$\uhp$
in~$\proj\CC$. Both Poisson transforms are given by integration
against the kernel function $R(\dv;z)^{1-s}$, where
$R(t;z) = \im(z)/(t-\nobreak z)(t-\nobreak \bar z)$, the integration
being over $\proj\RR$ for the usual Poisson transformation, and from
$z$ to $\bar z$ for the transverse one. We also need the inverse of
the Poisson transform. It can be given explicitly by integration of
the differential form $[u,R(t;\dv)^s]$, where
$[u,v]=\frac{\p u}{\p z} v \, dz 
+ u \frac{\p v}{\p\bar z} \, d\bar z$ is the
\indlbl{Gsfi}{Green's form}\emph{Green's form}, already used in
\cite{LZ}, which is closed if $u$ and $v$ are eigenfunctions of~$\Dt$
with the same eigenvalue.

These facts are reviewed in Chapter~\ref{ch-I}. See \S\ref{sssect-Grf}
for the Green's form and Section~\ref{ssect-prs} for the principal
series. The Poisson transformation is recalled in \S\ref{sssect-Pm},
and the transverse Poisson transformation is defined in
\S\ref{ssect-tPm}.
\medskip

In Chapter \ref{ch-II} we suppose that the discrete subgroup
$\Gm\subset G$ is cocompact. Here, $\Mf_s^0(\Gm) $ is just the space
$\E_s^\Gm$ of all $\Gm$-invariant solutions of
$\Dt u = s(1-\nobreak s)u$. Our first main result relates it to
cohomology groups with values in the spaces of analytic,
infinitely-often, and finitely-often differentiable functions
in~$\V{}s$:
\begin{mainthm}\label{thmA}For cocompact $\Gm \subset G$ and
$s\in \CC$, $0<\re s<1$, the space $\E_s^\Gm$ is canonically
isomorphic to the cohomology groups $H^1(\Gm;\V\om s)$,
$H^1(\Gm;\V\infty s)$, and $H^1(\Gm;\V p s)$ for $p\in \NN$,
$p\geq 2$.

To describe this isomorphism we associate to a given Maass form $u$
the analytic cocycle
\be\label{co}
 r_\gm(t) \= \int^{z_0}_{\gm^{-1}z_0} \left[ u,
 R(t;\dv)^s\right]\,,\ee
depending on a base point $z_0\in \uhp$. In the other direction, the
value of the Maass form $u(z)$ associated to a given analytic cocycle
$\{\ph_\gm\}$ is given in any compact subset of $\uhp$ by an explicit
finite sum of terms of the form $\tpoiss s(\ph_\gm)\,|\,\gm'$ with
$\gm,\,\gm'\in\Gm$.
\end{mainthm}
Bunke and Olbrich, \cite{BO95}, \cite{BO98}, proved that
$\E_s^\Gm \cong H^1(\Gm;\V\om s) \cong H^1(\Gm;\V\infty s)$ in a more
general setting (for automorphic forms on rank~$1$ symmetric spaces
and torsion-free discrete cocompact groups). Our approach is more
concrete and gives the isomorphism $\E_s^\Gm \cong H^1(\Gm;\V\om s)$
much more explicitly. The integral in \eqref{co} gives the map from
Maass forms to cohomology. For the map from cohomology to Maass
forms, the starting point is the model
\indlbl{Womi}{$\W\om s$}$\W\om s$ of the principal series in the
solutions of $\Dt u = s(1-\nobreak s)u$ near the boundary. We use a
space
\indlbl{Gomi}{$\G^\om_s$}$\G^\om_s$ of functions on the whole of
$\uhp$ such that $\G^\om_s \lra \W\om s$ is surjective. The kernel
\indlbl{Nomi}{$\N^\om$}$\N^\om$ of this morphism consists of compactly
supported functions. A 1-cocycle on~$\Gm$ with values in $\W\om s$
gives rise to a $2$-cochain with values in $\N^\om$. Evaluation of
this $2$-cochain on a $2$-cycle that represents the fundamental class
in $H_2(\Gm;\ZZ)$ provides us with an element $f_0\in\N^\om$, which
is unique up to linear combinations of the form
$f\,|\,(1-\nobreak\gm)$ with $f\in \N^\om$ and $\gm\in \Gm$. The
locally finite sum $u(z) = \sum_{\gm\in \Gm} f_0(\gm z)$ is
independent of all choices, and is the Maass form we looked for.

The construction of maps in both directions between $\E_s^\Gm $ and
$ H^1(\Gm;\V\om s) $ is the main result of Chapter~\ref{ch-II}.
\medskip

Chapter~\ref{ch-III} presents results concerning $H^1( \Dt; \V\om s)$
and related cohomology groups for infinite cyclic subgroups
$ \Dt\subset \PSL_2(\RR)$. It turns out, for instance, that the
restriction map from $H^1( \Gm;\V\om s)$ to $H^1( \Dt; \V\om s)$ in
the theorem above, where $\Dt$ is the subgroup generated by any
element $\gm_0\in \Gm$ of infinite order, is injective, so that a
Maass wave form $u\in \E_s^\Gm$ can be reconstructed from the single
element $r_{\gm_0}\in \V\om s$, without knowing the rest of the
cocycle. The results for the case that $\Dt$ is generated by a
parabolic (rather than a hyperbolic)
element of $\PSL_2(\RR)$ are used in the following chapter.
\medskip

In Chapter~\ref{ch-IV} we consider groups $\Gm$ with cusps. Here the
spaces
\indlbl{Moi}{$\Mf^0_s(\Gm)$}$\Mf^0_s(\Gm) \subset \Mf_s(\Gm)  
\subset \E_s^\Gm$ of, respectively,
\indlbl{Mcfii}{Maass cusp form}Maass cusp forms,
\indlbl{Mfii}{Maass form}Maass forms (at most polynomial growth), and
arbitrary
\indlbl{iei}{invariant eigenfunction}invariant eigenfunctions, are all
different. The dimension of $\E_s^\Gm$ is infinite, while the other
two are finite dimensional. The approach used for cocompact groups
has to be modified in several ways, as follows.

We have to look at more general cohomology groups. For
$\Gm$-modules~$V$, the
\indlbl{pcgi}{parabolic cohomology group}\emph{parabolic cohomology
group} $\Hpar^1(\Gm;V)\subset H^1(\Gm;V)$ is given by cocycles
$\{\ps_\gm\}$ that are of the form $\ps_\pi = a_\pi\,|\,\pi - a_\pi$
for all parabolic $\pi \in \Gm$, with $a_\pi \in V$. For the
\indlbl{mpcgi}{mixed parabolic cohomology group}\emph{mixed parabolic
cohomology group} $\Hpar^1(\Gm;V,W)$ the cocycle $\{\ps_\gm\}$ has
values in~$V$ and the $a_\pi$ are in a $\Gm$-module $W\supset V$.

The example of the period functions for $\Gmod=\PSL_2(\ZZ)$ leads us
to the space
\indlbl{Vsai}{$\V\sa s$}$\V\sa s$ of
\indlbl{savi}{semi-analytic vectors}``semi-analytic vectors'' in the
principal series. This is a $\Gm$-module satisfying
$\V\om s \subset \V\sa s \subset \V\infty s$, consisting, in the
standard model of the principal series representation
(functions on $\proj\RR$), of smooth
($C^\infty$) functions on~$\proj\RR$ that are real-analytic except for
finitely many points.

With these modifications one has the following analogue of
Theorem~\ref{thmA} for cusp forms on non-cocompact groups:
\begin{mainthm}\label{thmB}For cofinite discrete subgroups
$\Gm\subset \PSL_2(\RR)$ and $0<\re s<1$, the spaces $\Mf_s^0(\Gm) $,
$\Hpar^1(\Gm;\V\om s,\V\sa s)$, $\Hpar^1(\Gm;\V\sa s)$,
$\Hpar^1(\Gm;\V\infty s)$, and $\Hpar^1(\Gm;\V p s)$ with $p\in \NN$,
$p\geq 3$, are canonically isomorphic. The relation between Maass
cusp forms and the associated analytic cocycle has the same structure
as in Theorem~\ref{thmA}.
\end{mainthm}

\sssect{Example} In the case
\indlbl{mgi}{modular group}$\Gmod=\PSL_2(\ZZ)$, the cocycle determined
by \eqref{ancoc} represents a class in the mixed parabolic cohomology
group $\Hpar^1(\Gmod;\V\om s,\V\sa s)$: its values are in $\V\om s$,
and its value on the parabolic generator $T=\pm\rmatc1101 \in \Gmod$
is of the form $h\,|\,T-h$, with $h\in \V\sa s$. The
\indlbl{pfi}{period function}period function~$\ps$, on the other hand,
is related to a class in $\Hpar^1(\Gm;\V\sa s)$. It determines a
cocycle with values in $\V\sa s$, defined on the standard generators
by
\be \pm\rmatr0{-1}10 \mapsto \begin{cases}
\ps(x) &\text{ if }x>0\,,\\
-|x|^{-2s} \ps(-1/x)&\text{ if }x<0\,;
\end{cases}\qquad \pm\rmatc1101 \mapsto 0\,.
\ee
This cocycle vanishes on the parabolic element $\pm\rmatc1101$.
Finally, the last isomorphism in Theorem~\ref{thmB}, applied to the
modular group, implies that any $C^ \infty$-function on $(0,\infty)$
satisfying \eqref{3T} and the growth conditions given there is in
fact real-analytic, giving a strengthening of the main result
of~\cite{LZ}.\medskip

We prove most of the isomorphisms in Theorem~\ref{thmB} in
Chapter~\ref{ch-IV}. The isomorphism with $\Hpar^1(\Gm;\V\infty s)$
and $\Hpar^1(\Gm;\V p s)$ is established in
Chapter~\ref{ch-V}.\smallskip

If $s\neq\frac12$, the correspondence between Maass forms and
cohomology classes in the Theorems \ref{thmA} and~\ref{thmB} can be
extended to the whole of $\E_s^\Gm$. To do this, we introduce two
further spaces
\indlbl{vssgi}{$\V\wdg s$, $\V\wdgcp s$}$\V\wdg s  \supset \V\om s$
and $\V\wdgcp s\supset\V\om s$. The first consists of functions on
$\proj\RR$ which are real analytic except for finitely many points
and have singularities of a special type (Definition~\ref{wdgdef})
at these points, and the second is the same except that the finitely
many singularities must all be at cusps. Then we have:
\begin{mainthm}\label{thmC}For cofinite discrete subgroups
$\Gm \subset\PSL_2(\RR)$ and $0<\re s<1$, $s\neq \frac12$, the spaces
$\E_s^\Gm$, $\Hpar^1(\Gm;\V\om s,\V\wdg s)$ and
$\Hpar^1(\Gm;\V\wdgcp s)$ are canonically isomorphic.
\end{mainthm}
For the modular group we show in Proposition~\ref{prop-Hpf} how
$\Hpar^1(\Gmod;\V\wdgcp s)$ can be described as a quotient of the
space of all holomorphic functions on $\CC\sm(-\infty,0]$ that
satisfy the \indlbl{ttei}{three term equation}three term
equation~\eqref{3T}. In Proposition~\ref{prop-notall} we show that
the mixed parabolic cohomology group
$\Hpar^1(\Gmod;\V\om s,\V\wdg s)$ is a genuine subspace of
$H^1(\Gmod;\V\om s)$. We discuss briefly a notion of ``quantum Maass
forms'' which provides us with a space of objects with a modular
flavor that corresponds bijectively with $H^1(\Gmod;\V\om s)$.

If $\Gm$ has no cusps then parabolic cohomology is standard
cohomology. Theorems \ref{thmB} and~\ref{thmC} give no more
information than Theorem~\ref{thmA} in the cocompact case.


Bunke and Olbrich have shown that the space $\Mf_s^0(\Gm)$ (which is
equal to $\E_s^\Gm$ for cocompact~$\Gm$) is isomorphic to
$H^1(\Gm;\V{-\infty}s)$, where
\indlbl{V-ii}{$\V{-\infty}s$}$\V{-\infty}s$ denotes the space of
distribution vectors in $\V{}s$. In Chapter~\ref{ch-VI} we give an
explicit realization of the isomorphism on the cocycle level and use
it to express the Petersson scalar product in cohomological terms
(Theorem~\ref{thm-Ppcp}).



\sssect{Holomorphic automorphic forms and Maass forms}
We have mentioned at the start of the introduction that the classical
theory of cohomology classes attached to holomorphic automorphic
forms has similarities to what we do in these notes. There are also
many differences, due in particular to the anomalous behavior of the
principal series representation of $\SL_2(\RR)$ when the spectral
parameter is an integer. We refer to Chap.~IV, \S2 of \cite{LZ} for a
discussion of the similarities and differences in the modular case.

One of the differences is that here we need
infinite-dimensional modules for the cohomology, in
contrast to the the spaces of polynomials in the classical theory.
Infinite-dimensional modules are also needed by Knopp \cite{Kn74},
for the cocycles attached to holomorphic cusp forms of arbitrary real
weight.


\sssect{Selberg zeta function and transfer operator} 
As is well known, the values of the spectral parameter $s$ for which
the space of Maass cusp forms is non-zero occur among the zeros of
the \emph{Selberg zeta function}. The relationship between this fact
and the functional equation~\eqref{BBB}, in the case of the full
modular group, can be seen using the transfer operator of Mayer
\cite{Ma91}: on the one hand, the Selberg zeta function can be
expressed as the product of Fredholm determinants built with this
operator, and on the other hand the solutions of~\eqref{BBB}, i.e.,
the period functions of even Maass forms, give rise to eigenfunctions
of this operator with eigenvalue $1$ (or $-1$ if the plus sign
in~\eqref{BBB} is changed to a minus sign, corresponding to odd Maass
wave forms). These connections are discussed in detail in \cite{LZ1},
\cite{LZ}
(Chap.~IV, \S3), and \cite{Za2}.

For a number of families of other Fuchsian groups,
including the congruence subgroups $\Gm_0(p)$ of the modular group
and the Hecke triangle groups,  M\"oller and Pohl [25] and Pohl
 [27, 28, 29] have obtained a similar relationship between the more
general period functions of Maass forms considered in these notes and
eigenfunctions of appropriate transfer operators. For the case of the
full modular group again, the paper [5] relates cocycles for the
group to the eigenfunctions of a transfer operator different from
Mayer's original one. It would be interesting to have a similar
relationship for general Fuchsian groups between the Selberg zeta
function, eigenfunctions of transfer operators and cohomology
classes.

\sssect{Acknowledgements} The preparation of these notes has taken
many years. The Max Planck Institute in Bonn and the Coll\`ege de
France in Paris have enabled us many times to work on it together.
The two first-named authors thank both institutions for this support
and the excellent working conditions that they provided.



\sssect{Notations and conventions} We work with the standing
assumption that
\indlbl{s}{$s$}$s\in \CC$ satisfies $0<\re s<1$, and use
\indlbl{lds}{$\ld_s$}$\ld_s = s(1-\nobreak s)$. By
\indlbl{NN}{$\NN$}$\NN$ we denote the set
$\left\{n\in \ZZ\;:\; n\geq 0\right\}$.


We denote by \indlbl{G}{$G$}$G$ the Lie group
$\PSL_2(\RR) = \SL_2(\RR)/\{\pm \Id\}$, and denote by
$\,\matc abcd\,$ the element
\indlbl{mat}{$\matc\ast\ast\ast\ast$}$\,\pm\rmatc abcd\,$ of $G$. We
shall use various right representations of $G$, and use
\indlbl{ra}{$|$}$g:v\mapsto v\,|\,g$ as a general notation for the
action of $G$ in a
\indlbl{rm}{right module}right $G$-module.


% now use the standard equation numbering
\renewcommand\theequation{\equationstandard}


\skipbetweenchapters
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%   Chapter 1   eigenfunction                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\sect{Eigenfunctions of the hyperbolic Laplace operator}\label{ch-I}
This chapter has a preliminary character. It discusses concepts and
results needed in the next chapters. In
Sections~\ref{sect-efhp}--\ref{sect-bgtPt} we recall results
concerning eigenfunctions of the Laplace operator and principal
series representations that we treat in more detail in~\cite{BLZ}.
The averaging operators in Section~\ref{ssect-av} form another
important tool used in these notes.




\subsection{Eigenfunctions on the hyperbolic plane}\label{sect-efhp}
Maass forms are functions on the hyperbolic plane that satisfy
$\Dt u = \ld_s u$ and are invariant under a group of transformations.
We define in this subsection the space of all such eigenfunctions of
the Laplace operator and introduce several related spaces. An
important result is Theorem~\ref{thm-psC}, which plays for
eigenfunctions of $\Dt$ the role of Cauchy's theorem for holomorphic
functions.

\subsubsection{The hyperbolic plane}\label{ssect-hp}

By \indlbl{HH}{$\H$}$\H$ we denote the
\indlbl{hp}{hyperbolic plane}hyperbolic plane. We use two realizations
as a subset of $\proj\CC$. The first is the
\indlbl{uhp}{upper half-plane model of hyperbolic plane; $\uhp$}upper
half-plane model $\uhp= \{ z=x+iy\;:\; y>0 \}$, the other the disk
model
\indlbl{disk}{disk model of hyperbolic plane;
$\disk$}$\disk=\left\{ w\in \CC\;:\; |w|<1\right\}$. In the upper
half-plane model,
\indlbl{geo}{geodesic}geodesics are Euclidean vertical half-lines and
Euclidean half-circles with their center on the real axis. In the
disk model, geodesics are given by Euclidean circles intersecting the
boundary
\indlbl{uc}{$\uc$}$\p\,\disk=\uc = \{\x\in \CC\;:\; |\x|=1\}$
orthogonally and Euclidean lines through~$0$. The real projective
line
\indlbl{prR}{projective line; $\proj\RR$}
$\proj\RR = \RR \cup\{\infty\}$ is the boundary of the upper
half-plane. See
\indind{$\rho$; hyperbolic distance; $\dist$; volume element; $d\mu$;
boundary of hyperbolic plane; isometry group;
$\Dt$}{uhp-disk}Table~\ref{uhp-disk} for a further comparison between
both models.
\begin{table}[ht]
\[ \renewcommand\arraystretch{1.6}
\begin{array}{|l|c|c|}\hline
\text{model of $\H$}
& \uhp & \disk \\ \hline
\text{coordinate}
& z= x+iy=i \frac{1+w}{1-w} & w = \frac{z-i}{z+i} \\ \hline
\text{Laplace operator }\Dt& - y^2\bigl( \p_x^2+\p_y^2\bigr)&
-\bigl(1-|w|^2\bigr)^2\p_w\p_{\bar w}
\\
& \quad = (z-\bar z)^2\p_z\p_{\bar z}&
\\ \hline
\text{infinitesimal distance}&y^{-1} \sqrt{(dx)^2+(dy)^2}&
\frac{2\sqrt{(d\,\re w)^2+(d\,\im w)^2}}{1-|w|^2}
\\ \hline
\text{ hyperbolic distance} \dist&&\\
\rho(\dv,\dv) = \cosh (\dist(\dv,\dv))& 1+\frac{|z-z'|^2}{2yy'}&
1+\frac{2|w-w'|^2}{(1-|w|^2)(1-|w'|^2)}
\\ \hline
\text{volume element }d\mu& \frac{dx\, dy}{y^2} & \frac{4\, d\,\re w\;
d\,\im w}{(1-|w|^2)^2}
\\ \hline
\hspace*{-.5em} {\renewcommand\arraystretch{.8}
\begin{array}{l}
\text{orientation preserving}\\
\text{isometry group}\end{array}}
& G=\PSL_2(\RR)& G\cong \mathrm{PSU}(1,1)
\\
\matc ABCD = \matr1{-i}1i \matc abcd \matr1{-i}1i^{-1}
& \matc abcd : z\mapsto \frac{az+b}{cz+d}& \matc A B {\bar B} {\bar
A}: w\mapsto \frac{ Aw+B}{\bar Bw+\bar A}
\\ \hline
\text{maximal compact subgroup }K& \mathrm{PSO}(2)& \mathrm{PSU}(1)
\\
\text{point fixed by $K$}& i& 0
\\ \hline
\text{boundary }\p\H& \p\uhp=\proj\RR & \p\disk=\uc
\\ \hline
\end{array}
\]
\caption{Upper half-plane model and disk model of the hyperbolic plane
$\H$.}\label{uhp-disk}
\end{table}

\sssect{The space of eigenfunctions} By
\indlbl{E}{$\E_s$}$\E_s$ we denote the space of solutions of
\be \label{Lapl-eq}
\Dt u = \ld_s\,u\text{ in }\H,\qquad \ld_s=s(1-s)\,.\ee
The \indlbl{Lo}{Laplace operator}Laplace operator
$\Dt=-y^2\p_x^2-y^2\p_y^2$ is an elliptic differential operator with
real-analytic coefficients. Hence all elements of $\E_s$ are
real-analytic functions. This operator commutes with the action of
the group $G$ (on the right) given by
\[(u\,|\,g)(z)=u(gz)\,.\]
(We will use \indlbl{zco}{$z$ coordinate on $\H$ and $\uhp$}$z$ to
denote the coordinate in both $\uhp$ and $\disk$ when we make
statements applying to both models of $\H$.)
Obviously, $\E_s=\E_{1-s}$. If $U$ is an open subset of $\H$, we
denote by
\indlbl{Esh}{$\E_s(\dv)$}$\E_s(U)$ the space of solutions of
$\Dt u=\ld_su$ on $U$, thus defining $\E_s$ as a sheaf on~$\H$. We
will refer to elements of $\E_s=\E_s(\H)$ and of $\E_s(U)$ as
\indlbl{ldsef}{$\ld_s$-eigenfunction}\emph{$\ld_s$-eigenfunctions
of~$\Dt$}.

So $\E_s$ denotes a sheaf as well as the space of global sections of
that sheaf. For other sheaves we will allow ourselves a similar
ambiguity.

\subsubsection{Examples}\label{ssect-exef}The functions
$i_{s,0}(z) = y^s$ on $\uhp$ is an element of $\E_s$ that occurs in
the constant term of Fourier expansions of Maass forms. That term is
a linear combination of $i_{s,0}$ and $i_{1-s,0}$, or of $i_{1/2,0}$
and $\ell_{1/2,0}(z) = y^{1/2}\ln y$ if $s=\frac12$. This function
$\ell_{1/2,0}$ is the value at $s=\frac12$ of the family
\indlbl{ell-h}{$\ell_{s,0}$}$\ell_{s,0} 
= \frac1{2s-1}\left( i_{s,0}-i_{1-s,0}\right)$ of $N$-invariant
elements of $\E_s$. The other terms of those Fourier expansions may
contain the following elements of~$\E_s$:\indind{$i_{s,\al}$,
$k_{s,\al}$}{ikdef}
\be\label{ikdef}
\begin{aligned}
k_{s,\al}(z) &\= \sqrt y \, K_{s-1/2}(|\al|y)\, e^{i\al x}\,,
\\
i_{s,\al}(z)&\= \frac{\Gf\bigl(s+\frac12\bigr)}{|\al/2|^{s-1/2}}\,
\sqrt y \, I_{s-1/2}(|\al|y)\, e^{i\al x}\,,
\end{aligned}
\ee
for $\al\in \RR\sm\{0\}$, with the modified Bessel functions
\indlbl{IK-Bessel}{$I_\mu(\dv)$, $K_\mu(\dv)$}$I_\mu(\dv)$ and
$K_\mu(\dv)$. These functions on $\uhp$ transform according to the
character $\matc 1x01\mapsto e^{i\al x}$ of
\indlbl{Ndef}{$N\;\subset\;G$}$N=\bigl\{\matc1\ast01\bigr\}\subset G$.

The group
\indlbl{Kdef}{$K\;\subset\;G$}$K=\mathrm{PSO}(2) \subset G$ has
characters
$\matr{\cos\th}{\sin\th}{-\sin\th}{\cos\th}\mapsto e^{2in\th}$ with
$n\in \ZZ$. Functions transforming according to such a character are
easiest described in the disk model, with Legendre functions or with
hypergeometric
functions:\indind{$P_{s,n}$}{pn-def}\indind{$Q_{s,n}$}{qn-def}
\beqabclbl{pqsndef}
\begin{align}\label{psn-def}
&\begin{aligned}
P_{s,n}(re^{i\th}) &\= P^n_{\!s-1} \Bigl( \frac{1+r^2}{1-r^2}\Bigr)\,
e^{in\th}\\
&\= \frac{\Gf(s+n)}{|n|!\; \Gf(s-|n|)} \, r^{|n|}\,
\hypg21{\Bigl(1-s,s;1+|n|;\frac{r^2}{r^2-1}\Bigr)}\,,
\end{aligned}\displaybreak[0]\\
\label{qsn-def}
&\begin{aligned}
Q_{s,n}(re^{i\th})&\= Q^n_{s-1} \Bigl( \frac{1+r^2}{1-r^2}\Bigr)\,
e^{in\th}\\
&\= \frac{(-1)^n}2\, \frac{\Gf(s)\,\Gf(s+n)}{\Gf(2s)}\, r^{-n}\,
(1-r^2)^s\, \hypg21{\bigl(s-n,s;2s;1-r^2\bigr)}\,.
\end{aligned}
\end{align}
\eeqabc
(Note the shift in the spectral parameter in $P^n_{\!s-1}$ and
$P_{\!s,n}$.)
We have $P_{\!s,n}=P_{\!1-s,n}\in \E_s$ and
$Q_{s,n}\in\E_s(\disk\sm\{0\})$.

Section~A.1.3 in the appendix of~\cite{BLZ} gives also formulas for
elements of $\E_s$ that transform according to a character of the
group
\indlbl{Adef}{$A\;\subset\;G$}$A=
\big\{ \matc{y^{1/2}}00{y^{-1/2}}\;:\; y>0\bigr\}\subset G$. Of these
functions we will use $f^R_{s,\al}$ and $f^L_{s,\al}$.
See~\eqref{frt}.\smallskip

Starting from the $\ld_s$-eigenfunctions of~$\Dt$ considered up till
now, we can produce other ones by translating them. If
$g \cdot 0 = w'$ for $g\in G$ and $w'\in \disk$, then $r=|g^{-1}w|$
satisfies $\frac{1+r^2}{1-r^2} = \rho(w,w')$, with $\rho$ as in
Table~\ref{uhp-disk}. The functions \indind{$p_s(\dv,\dv)$;
$q_s(\dv,\dv)$}{pqs-def}
\be\label{pqs-def}
\begin{aligned}
p_s(w,w') &\= P_{\!s,0}(g^{-1}w) \= P^0_{\!s-1}
\bigl(\rho(w,w')\bigr)\,,\\
q_s(w,w') &\= Q_{s,0}(g^{-1}w) \= Q^0_{s-1}\bigl(\rho(w,w')\bigr)\,,
\end{aligned}
\ee
are \indlbl{ppi}{point-pair invariant}\emph{point-pair invariants},
i.e., they depend only on the hyperbolic distance between $w$ and
$w'$. Hence they are symmetric in $w$ and~$w'$, and satisfy for all
$g\in G$:
\be\label{pq-Ginv}
p_s(g z,g z') = p_s(z,z')\,,\qquad q_s(g z,g z') = q_s(z,z')\,. \ee
 They are $\ld_s$-eigenfunctions of $\Dt$ in both variables. One calls
 $q_s$ the
 \indlbl{frk}{freespace resolvent kernel}\emph{free-space resolvent
 kernel}. We have $p_s(\dv,w') \in \E_s$, and
$q_s(\dv,w') \in \E_s(\disk\sm\{w'\})$ for fixed $w'\in \disk$.
\smallskip

Shifting $i_{s,0}:z\mapsto y^s$ by $\matr01{-1}t$, with $t\in \RR$,
gives
\indind{$R(\dv;\dv)^s$}{R}
\be\label{R-def}
R(t;z)^s \= \Bigl( \im \frac1{t-z} \Bigr)^s \=
\frac{y^s}{|t-z|^{2s}}\,. \ee
We have $R(t;\dv)^s  \in \E_s $. Moving $t$ off the real line gives
rise to a multivalued function
\be R(\z;z)^s \= \frac{y^s}{(\z-z)^s(\z-\bar z)^s}\,. \ee
For $\z\in \CC$ it is an element of $\E_s(U)$ for simply connected
$U\subset\uhp$ not containing $\z$ or~$\bar\z$. We choose the branch
such that $\arg(\z-\nobreak z) + \arg(\z-\nobreak \bar z)=0$ for
$\z\in \RR$.

For $\al\in \RR$ and $\re s>\frac12$, we can integrate
$e^{i\al t}R(t;z)^s$ over~$\RR$ to obtain
\be\label{Kint}
e^{i\al x} \int_{-\infty}^\infty e^{i\al t} \frac{y^s\,
dt}{(y^2+t^2)^s} \=
\begin{cases}
\sqrt\pi \, \frac{\Gf(s-\frac12)}{\Gf(s)} \,
i_{1-s,0}(z)&\quad(\al=0)\,,\\
\sqrt\pi \, 2^{3/2-s} \, \frac{|\al|^{s-1/2}}{\Gf(s)} \,
 k_{s,\al}(z)&\quad(\al\neq0)\,.
\end{cases}\ee
This continues meromorphically, holomorphically in $s$ if $\al\neq0$,
and having in the region $\re s>0$ only a first order singularity at
$s=\frac12$ if $\al=0$.

\subsubsection{Green's form}\label{sssect-Grf}
\indlbl{Gf}{Green's form} We recall the bracket operation from
\cite{LZ}, already mentioned in the introduction. There
are two versions, which differ by an exact form:
\indind{$[\dv,\dv]$; $\{\dv,\dv\}$}{[]def}
\be\label{[]def}
[u,v]\=u_z\,v\,dz\,+\,u\,v_{\bar z}\,d\bar
z\,,\qquad\{u,v\}\=2i\,[u,v]\,-\,i\,d(uv)\,.\ee
These formulas make sense in both the upper half-plane and the disk
model of~$\H$, and have the properties
\beqabc
\begin{align}
\label{[]invar}
[u\circ g, v \circ g] &\= [u,v]\circ g\quad\text{ for each }g\in G\,,
\displaybreak[0]\\
\label{som[]}
[u,v]+[v,u]&\= d(uv)\,,
\displaybreak[0]\\
\label{d[]}
d[u,v]&\= \frac1{2i} (u \, \Dt v - v \, \Dt u) \, d\mu\,,
\displaybreak[0]\\
\label{{}alt}
\{v,u\}&\=-\{u,v\}\,.
\end{align}
\eeqabc
So $[u,v]$ is a closed $1$-form on~$U$ if $u,v\in \E_s(U)$ for an open
$U\subset\H$.



The bracket with $q_s$ gives for elements of $\E_s$ a substitute for
Cauchy's theorem:
\begin{thm}\label{thm-psC}Let $C$ be a piecewise smooth simple closed
curve in~$\H$ and $u$ an element of $\E_s(U)$, where $U\subset\H$ is
some open set containing $C$ and its interior. Then for $w\in\H\sm C$
we have
\be\label{psCc} \int_C\left[ u, q_s(\dv,w)\right] \= \begin{cases}
\pi i \, u(w)&\text{ if $\,w\,$ is inside $C$},\\ \;0&\text{ if
$\,w\,$ is outside $C$,} \end{cases}\ee
where the curve $C$ is traversed in the positive direction.
\end{thm}
See Theorem~2.1 in~\cite{BLZ}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Principal series}\label{ssect-prs} All the coefficient
modules used in the cohomology groups mentioned in the introduction
are spaces of vectors in the principal series representation
associated to the spectral parameter~$s$\indlbl{spr}{spectral
parameter; $s$}. The standard realizations of the principal series
representation use spaces of functions on the boundary $\p\H$ of the
hyperbolic plane. With the Poisson transform we can also use a
realization in $\E_s$.

We write \indlbl{Vs}{$\V{}s$}$\V{}s$ to denote ``the''
\indlbl{ps}{principal series}principal series representation when we
do not want to specify precisely the space under consideration.
Spaces $\V\infty s$ and $\V\om s$ of smooth and analytic vectors are
identified with the appropriate superscript.

In \cite{BLZ} we treat the material in this section in more depth. In
particular, we study the various models more systematically. Each of
the models of $\V{}s$ has its advantages and disadvantages.

\subsubsection{Models of the principal series on the boundary of the
hyperbolic plane}\label{ssect-mps-bd}We list some standard models of
the principal series.

\sssect{Line model}In the introduction we already mentioned the well
known model of $\V{}s$, consisting of functions on $\RR$ with the
transformation behavior
\indind{$|_{2s}$}{|s}
\be\label{tr-lm} \ph\bigl|_{2s}\,\matc abcd
(x) \= |cx+d|^{-2s} \ph\Bigl(\frac{ax+b}{cx+d}\Bigr)
\ee
under $\matc abcd\in G$. To get a sensible result at $x=-\frac dc$, we
need to require that $\ph$ behaves well as $|x|\rightarrow\infty$. By
\indlbl{Vsoi}{smooth vectors in principal series;
$\V\infty s$}$\V\infty s$, the space of \emph{smooth vectors} in
$\V{}s$ we denote the space of $\ph \in C^\infty(\RR)$ that have an
expansion
\be\ph(t)\;\sim\;|t|^{-2s}\;\sum_{n=0}^\infty\,c_n\,t^{-n}\label{asy}\ee
as $|t|\to\infty$. Similarly, the space
\indlbl{anv}{analytic vectors in principal series; $\V\om s$}$\V\om s$
of \emph{analytic vectors} consists of the $\ph \in C^\om(\RR)$
\indlbl{Com}{$C^\om$}(real-analytic functions on~$\RR$) for which the
series appearing on the right-hand side of~\eqref{asy} converges to
$\ph(x)$ for $|x|\geq x_0$ for some~$x_0$. Analogously, we define
\indlbl{Vp}{$\V p s$}$\V p s$, $p\in \NN$, as the space of
$\ph \in C^p(\RR)$ satisfying \eqref{asy} with the asymptotic
expansion replaced by a Taylor expansion of order~$p$.

We call this the \indlbl{lm}{line model of principal series}\emph{line
model} of $\V{}s$. It is well known and has a simple transformation
formula \eqref{tr-lm} that reminds us of the transformation behavior
of holomorphic automorphic forms. It has the disadvantages that we
need to specify the behavior as $|x|\rightarrow\infty$ separately,
and that it requires some work to check that the spaces $\V\om s$,
$\V\infty s$, $\ldots$ are preserved under the action of~$G$.

Often we shall write $\ph\,|\,g$ instead of $\ph|_{2s}\,g$ if there is
no danger of confusion.

\sssect{Projective model}The relation
\indlbl{pm}{projective model of principal
series}\indind{$\ph^\prj$}{phprjd}
\be \label{phprjd}\ph^\prj(t) \= (1+t^2)^s \ph(t)
\ee
gives a model for which $\V\om s$, $V^\infty_s$, and the $\V p s$
correspond to respectively $C^\om(\proj\RR)$, $C^\infty(\proj\RR)$
and $C^p(\proj\RR)$ of respectively real analytic, smooth and $p$
times continuously differentiable functions on~$\proj\RR$. The action
of $G$ is described by the more complicated formula
\indind{$|_{2s}^\prj$}{|prj}
\begin{equation}\label{projac}
f\bigl|_{2s}^\prj\,\matc abcd\bigl(t\bigr)\=
\Bigl(\frac{t^2+1}{(at+b)^2+(ct+d)^2} \Bigr)^s
\,f\Bigr(\frac{at+b}{ct+d} \Bigl).
\end{equation}
The factor $ \bigl(\frac{t^2+1}{(at+b)^2+(ct+d)^2}\bigr)^s$ is
real-analytic on the whole of $\proj\RR$. Thus it is immediately
clear that the action of $G$ preserves real-analyticity, smoothness
and $p$ times continuous differentiability. A drawback is that the
point $i$, corresponding to the choice of $K$ as maximal compact
subgroup, plays a special role. In \S1.1, \cite{BLZ}, we mention the
plane model of the principal series that does not have this drawback.
\sssect{Circle model} \indlbl{cm}{circle model of principal series}
The circle model is directly related to the projective model by the
inverse transformations $\x=\frac{t-i}{t+i}$ and
$t=i\frac{1+\x}{1-\x}\,$, in $\proj\CC\,$, identifying the projective
line $\proj\RR$ to the unit circle $\uc$ in $\CC$. This leads to the
circle model of $\V{}s\,$, in which the action of
$g=\matc abcd \in \PSL_2(\RR)$ is described by
\indlbl{gtld}{$\tilde g$}$\tilde g = \matc1{-i}1i\,g\,\matc1{-i}1i^{-1}
=\matc AB{\bar B}{\bar A}$ in $\mathrm{PSU}(1,1)\subset\PSL_2(\CC)$,
with $A=\frac12(a+ib-ic+d),\;B=\frac12(a-ib-ic-d)\,$:
\indind{$|_{2s}^\ci$}{trfcimo}
\be\label{trfcimo}
f\bigl|_{2s}^\ci\,g\,\bigl(\x\bigl)
\=|A\x+B|^{-2s} f\Bigl(\frac{A\x+B}{\vphantom{A^A}\bar B\x+\bar
  A}\Bigr)\qquad(|\x|=1)\,.\ee
The factor $|A\x+B|$ is non-zero on the unit circle, since
$|A|^2-|B|^2=1$. The relation with the previous models is given by
\indind{$\ph^\ci$}{phci}
\be \label{phci}\ph^\ci(e^{-2i\th})\= \ph^\prj\bigl(\cot\th\bigr)
    \=|\sin\th|^{-2s}\,\ph(\cot\th)\,. \ee

\sssect{Realization of $\V\om s$ in holomorphic functions}The
restriction of a holomorphic function on a neighborhood of $\uc$ in
$\CC$ to $\uc$ is real-analytic, and since every real-analytic
function on $\uc$ is such a restriction, $C^\om(\uc)$ can be
identified with the space $\limind {} \hol(U)$, where $U$ in the
inductive limit runs over all open neighborhoods of $\uc$ and where
\indlbl{hol}{$\hol(\dv)$}$\hol(U)$ denotes the space of holomorphic
functions on~$U$. One can rewrite the automorphy factor in
\eqref{trfcimo} as
$\left( ( \bar A+\bar B \x)(A + B \x^{-1}) \right)^{-s}$, which is
holomorphic near $\uc$. It can be extended to a holomorphic and
one-valued function on a neighborhood of $\uc$ in $\proj\CC$, in
fact, outside a path from $0$ to $-B/A$ and a path from $\infty$ to
$-\bar A/\bar B$. In other words, in the description of $\V\om s$ as
$\limind U \hol(U)$, the action of $G$
becomes\indind{$\ph|_{2s}^\ci$}{phia-h}
\be \label{phia-h} \ph|_{2s}^\ci\, g\,\,(w)\=\bigl[(\bar A+\bar
Bw)(A+B/w)\bigr]^{-s} \,\ph(\tilde g w)\;.
\ee

In the projective model, we have similar descriptions. Now $U$ runs
through neighborhoods of $\proj\RR$ in $\proj\CC$. The action
\eqref{projac} can be rewritten as\indind{$|_{2s}^\prj$}{grotesque}
\begin{equation}\label{grotesque}
f|^\prj_{2s}\, \matc abcd (z) \= \bigl(a^2+c^2\bigr)^{-s} \, \Bigl(
 \frac{z-i}{z-g^{-1}(i)} \Bigr)^s \, \Bigl(
\frac{z+i}{z-g^{-1}(-i)}\Bigr)^s \, f\Bigl(\frac{az+b}{cz+d}\Bigr) .
\end{equation}
This unwieldy formula shows that the automorphy factor is holomorphic
on $\proj\CC$ minus a path from~$i$ to $g^{-1}i$ and a path from $-i$
to $g^{-1}(-i)$.

\sssect{Topology} We have not yet discussed topologies on the spaces
in $\V{}s$. For the cohomology groups, we will use $\V\om s$,
$\V\infty s$, and other spaces in $\V{}s$, algebraically.

The natural topology on $\V p s$ is given by the finitely many
seminorms $\|\ph\|_j$, $0\leq j \leq p$, where
\indlbl{snj}{seminorms on $\V p s$ and $\V\infty s$; $\|\dv\|_j$}
\be\label{seminorm} \|\ph\|_j \= \sup_{x\in \p\H} \bigl| \ph|_{2s}
\,{\bold W}^j
(x)\bigr|\,, \ee
and where $\bold W=\matr01{-1}0$ in the Lie algebra of~$G$. By
$\ph\mapsto\ph|_{2s} \,{\bold W}$ we denote the corresponding action
in $\V{}s$. In the circle model, $\bold W$ corresponds to the
differential operator $2i\,\x\,\p_\x$, in the projective model to
$(1+t^2) \, \p_t$, and in the line model to
$(1+\nobreak x^2) \, \p_x+2s\,x$. The natural topology on the space
$\V\infty s=\bigcap_{p\in \NN} \V p s$ is given by the collection of
all seminorms $\|\dv\|_p$, $p\in \NN$.


The topology on $\V\om s \cong\limind {} \hol(U)$ can be defined as
the inductive limit topology given by the supremum norms on the
sets~$U$. The inclusion $\V\om s\lra \V\infty s$ is continuous with
dense image. With these topologies, $\V\om s$ and $\V\infty s$ are
irreducible continuous representations of $G$. Here the restriction
$0<\re s<1$ is essential. Irreducibility does not hold when
$s\in \ZZ$.

\sssect{Hyperfunctions}We put
\indlbl{HHH}{$\HHH$}
\be \HHH(\uc)\=\limind U\hol(U\sm\uc)\,,\qquad \HHH(\proj\RR) \=
\limind V \hol(V\sm\proj\RR)\ee
where $U$ runs over the neighborhoods of $\uc$ in $\CC$ and $V$ over
the neighborhoods of $\proj\RR$ in $\proj\CC$. The spaces
\indlbl{hfs}{hyperfunction; $C^{-\om}$}$C^{-\om}(\uc)$ and
$C^{-\om}(\proj\RR)$ of \emph{hyperfunctions} on $\uc$, respectively
$\proj\RR$, are the quotients in the exact sequences
\be\label{hf-def}
\begin{aligned}0\lra C^\om(\uc)
&\lra\HHH(\uc)\lra\hpf(\uc)\lra 0\,,\\
0\lra C^\om(\proj\RR)
&\lra\HHH(\proj\RR)\lra\hpf(\proj\RR)\lra 0\,.
\end{aligned}
\ee
See, e.g., \S1.1 of~\cite{Skr}. Actually, the quotients
$\hol( U \sm \uc)/\hol(U)$ and $\hol(V \sm\proj\RR) /\hol(V)$ do not
depend on the choice of $U$, respectively $V$, so they give models
 for $\hpf(\uc)$ and $\hpf(\proj\RR)$ for any choice of $U$,
respectively~$V$. Intuitively, a hyperfunction is the jump
across~$\uc$, respectively~$\proj\RR$.

The actions in \eqref{phia-h} and \eqref{grotesque} make sense on the
spaces of holomorphic functions in deleted neighborhoods
$\hol(U \sm \uc)$ and $\hol(V\sm \proj\RR)$. This gives an action of
$G$ on $\HHH(\uc)$ and $\HHH(\proj\RR)$, and hence on the
corresponding spaces of hyperfunctions. We call $\HHH_s$ the space
$\HHH$ with this action, in the realizations $\HHH(\uc)$ and
$\HHH(\proj\RR)$, and
\indlbl{V-om}{$\V{-\om}s$}$\V{-\om}s$ the resulting representation
of~$G$ in the hyperfunctions. Thus we have an exact sequence of
$G$-modules
\be \label{hdefG}
0 \lra \V\om s \lra \HHH_s \lra \V{-\om}s
\lra 0\,, \ee
realized in the circle model and in the projective model. The line
model is inconvenient for hyperfunctions.

We can embed the space $C^\om(\uc)$ of analytic functions on $\uc$ in
the following way:
\be \ph \in \hol(U) \quad\mapsto\quad w\mapsto\begin{cases}
\ph(w)&\text{ if }w\in U,\, |w|<1\,,\\
0&\text{ if }w\in U,\, |w|>1\,.
\end{cases}
\ee
Let $[f]\in C^{-\om}(\uc)$ be the hyperfunction represented by
$f\in\hol(U\sm\ci^1)$. Then $[f]\in C^\om(\uc)$ if and only if the
restrictions of $f$ to $U\cap\{|w|<1\}$ and $U\cap\{|w|>1\}$ both
extend holomorphically across the circle. In the projective model we
have a similar embedding.

\sssect{Duality} Let $\ph,\ps\in \HHH(\uc)$ be represented by
$f,\, h\in \hol(U\sm\uc)$ for some neighborhood $U$ of~$\uc$. There
is an annulus $e^{-a} \leq |w| \leq e^a$ contained in~$U$. Let
\indlbl{C+C-}{$C_+$, $C_-$}$C_+$ be a contour $|w|=c_+ \in [e^{-a},1)$
encircling $0$ once in the positive direction, and let $C_-$ be a
similar contour $|w|=c_-\in (1,e^a]$. Then the
integral\indlbl{du}{duality}\indind{$\langle\dv,\dv\rangle$}{dual-ci}
\be\label{dual-ci}
\langle\ph,\ps\rangle\= \frac1{2\pi i} \,\biggl(
\int_{C_+}-\int_{C_-}\biggr) \, f(w)h(w)\,\frac {dw}w \ee
is independent of the choice of the contours, as long as they are
continuously deformed within $U\sm \uc$. So the actual neighborhood
is not important. Moreover, if $f$ and $h$ are both in $\hol(U)$,
then Cauchy's theorem gives $\langle\ph,\ps\rangle=0$. Thus, we get
an induced pairing $C^\om(\uc) \times C^{-\om}(\uc) \lra \CC$, which
we denote also by $\langle\dv,\dv\rangle$.


The description in the projective model is
\twocolwithpictr{
\be
\begin{aligned} \langle \ph&,\ps\rangle \;=\; \frac1\pi \, \biggl(
\int_{C_+} - \int_{C_-} \biggr)\\&\quad\hbox{} \cdot\ph(z) \, g(z)\,
\frac{dz}{1+z^2}\,,
\end{aligned}\ee
where $\ph\in \hol(U)$ for some neighborhood $U$ of $\proj\RR$ in
$\proj\CC$, and $g \in \hol(U\sm\proj\RR)$ represents
$\ps\in C^{-\om}(\proj\RR)$. The contours $C_+\subset \uhp \cap U$
and $C_-\subset \lhp\cap U$ are homotopic with $\proj\RR$. The
orientation in $\CC$ of $C_+$ is positive and the orientation of
$C_-$ negative. It turns out that for all $g\in G$}{
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\be \left\langle \ph|_{2-2s}\, g, \,\ps|_{2s}\, g \right\rangle \=
\langle \ph,\ps\rangle\,. \ee
Thus, we have a bilinear invariant pairing
$\V\om{1-s} \times\V{-\om}s\lra \CC$.

{}From \eqref{dual-ci} we see that for fixed $h\in \hol(U\sm\uc)$ the
map $f\mapsto \langle f,h\rangle$ is continuous with respect to the
supremum norm of~$f$ on~$U$. Thus $\ph \mapsto \langle\ph,\ps\rangle$
is a continuous linear form on $\V\om{1-s}$ for fixed
$\ps\in \V{-\om}s$. It turns out that this gives an
\label{identification}identification of $\V{-\om}s$ with the
continuous dual of $\V\om{1-s}$.

It may happen that $\ph\mapsto \langle\ph,\ps\rangle$ can be
continuously extended to $\V\infty s$ for the topology
on~$\V\infty s$. Such linear forms are \indlbl{di}{distribution
vector in principal series; $\V{-\infty}s$}distribution vectors
in~$\V{}s$. The space $\V{-\infty}s$ consists of the distribution
subspace of~$\V{-\om}s$. It can be identified with the continuous
dual of $\V\infty{1-s}$.
\sssect{Basis}There are elements $\eph_{s,n}\in \V\om s$, $n\in \ZZ$,
such that $\langle \eph_{1-s,n},\eph_{s,m}\rangle =
\dt_{n,-m}$:\indind{$\eph_{s,n}$, $\eph_{s,n}^\prj$,
$\eph_{s,n}^\ci$}{phisn}
\beqabclbl{phisn}
\begin{align}\label{phisndef}
  \eph_{s,n}(t)&\=(t^2+1)^{-s} \, \Bigl( \frac{t-i}{t+i}\Bigr)^n\,,
\displaybreak[0]\\
\label{phisndef1}
  \eph^\prj_{s,n}(t) &\= \Bigl( \frac{t-i}{t+i}\Bigr)^n\,,
\displaybreak[0]\\
  \label{phisndef2} \eph^\ci_{s,n}(\x)
  &\=\x^n\,.
\end{align}\eeqabc
Fourier theory gives an expansion
$\ph=\sum_{n=-\infty}^\infty c_n  \, \eph_{s,n}$ of each element
$\ph\in\V{-\om}s$. We have
\be\label{coi}\begin{aligned}
\V\om s &\= \bigl\{ \sum c_n \, \eph_{s,n} \;:\; c_n = \oh\bigl(
e^{-a|n|}\bigr)\text{ for some }a>0\bigr\},\\
\V\infty s &\= \bigl\{ \sum c_n \, \eph_{s,n} \;:\; c_n =
\oh\bigl((1+|n|)^{-A}\bigr)
 \text{ for all }A>0\bigl\}\,,\\
\V{-\infty}s&\=\bigl\{\sum c_n \, \eph_{s,n} \;:\; c_n = \oh \bigl(
(1+|n|)^a\bigr)\text{ for some }a>0\bigr\}\,,\\
 \V{-\om}s &\= \bigl\{ \sum c_n \, \eph_{s,n} \;:\;
 c_n=\oh\bigl(e^{A|n|}\bigr)
   \text{ for all }A>0\bigr\}\,.
\end{aligned}\ee


\sssect{Isomorphism}For $0<\re s < 1$, the $G$-modules $\V{-\om}s$ and
$\V{-\om}{1-s}$ are isomorphic. The intertwining operator
\indlbl{I}{intertwining operator;
$I_s$}$I_s : \V{-\om}s\lra \V{-\om}{1-s}$ can be given on the basis
vectors in \eqref{phisn}:
\be \label{Idef}
I_s \eph_{s,n} \= \frac{\Gf(s) \, \Gf(1-s+n)}{\Gf(1-s) \, \Gf(s+n)} \,
\eph_{1-s,n} \,. \ee


\sssect{Sheaves}The definitions of $\V\om s$, $\V\infty s$ and the
$\V p s$, are local. We can form the corresponding sheaves. We shall
use this often for $\V\om s$. We formulate this for the projective
model and leave the analogous definitions in the other models to the
reader.

For each open set $I\subset \proj\RR$, we
define\indind{$\V\om s(I)$}{Vsheaf}
\be\label{Vsheaf}
\V\om s(I) \= \limind U \hol(U)\,, \ee
where $U$ runs through the neighborhoods of $I$ in $\proj\CC$. Note
that we allow ourselves to write $\V\om s$ instead of
$\V\om s(\proj\RR)$.

We also use the notation\indind{$\V\om s[F]$}{}
\be \label{V[]}
\V\om s[F] \= \V\om s \left( \proj\RR \sm F\right)
\ee
where $F$ is a finite subset of $\proj\RR$. We will simply write
$\V\om s[\x_1,\ldots,\x_n]$ instead of
$\V\om s[\{\x_1,\ldots,\x_n\}]$. If we impose a condition on the
sections of $\V\om s$ at the points $\x_j$ we write
$\V{\om,\mathrm{cond}}s[\x_1,\ldots,\x_n]$. For instance,
$\ph \in \V{\om,\infty}s[\x_1,\ldots,\x_n]$ is an element of
$\V\infty s$ with analytic restriction to
$\proj\RR \setminus\{\x_1,\ldots,\x_n\}$.


The $G$-module $\V\om s$ is naturally included in the
$G$-module\indind{$\V\fs s$}{}
\be\label{Vfs}
 \V\fs s \= \limind F \V\om s\left( \proj\RR \sm F \right)\,,\ee
where $F$ runs through the finite subsets of $\proj\RR$. So $\V\fs s$
can be viewed as the union of all $\V\om s[F]$, with identification
of $f\in \V\om s[F_1]$ with its image in $\V\om s[F]$ if
$F_1\subset F$. The space $\V\fs s$ is not a subspace of $\V{-\om}s$.
We call $\V\fs s$ the space of
\indlbl{sav}{semi-analytic vector in principal
series}\emph{semi-analytic vectors} in the principal series
representation. With an additional condition we write
\be\label{Vfs-cond} \V{\fs,\mathrm{cond}}s \= \limind F
\V{\om,\mathrm{cond}}s \bigl( \proj\RR\sm F \bigr)\,. \ee

By \indlbl{Vbs}{singularity; $\bsing$}$\bsing(f)$ for $f\in \V\fs s$
we denote the minimal finite set
$\{\x_1,\ldots,\x_n\}\subset\proj\RR$ such that
$f\in \V\om s[\x_1,\ldots,\x_n]$. We call these $\x_j$ the
\emph{singularities} of~$f$.
\medskip

\sssect{Terminology} Usually one denotes by $\V p s$ the space of $p$
times differentiable vectors in a Hilbert space $\V{L^2} s$ in
$\V{}s$, for which the $\eph_{s,n}$ form a complete orthogonal
system. We use $\V p s$ to denote \emph{functions} that are $p$ times
continuously differentiable in the projective of circle model. This
space is smaller than the space of $p$ times differentiable vectors
in~$\V{L^2}s$.

Our spaces $\V\om s$, $\V\infty s$ coincide with the spaces of
analytic and smooth vectors in $\V{L^2}s$, and similarly for
$\V{-\infty}s$ and $\V{-\om}s$. It seems hard and hardly interesting
to characterize spaces like $\V\fs s$ in terms of the Hilbert
space~$\V{L^2}s$. To summarize: our upper indices in $\V\ast s$ refer
to the behavior of \emph{functions} in the circle and projective
model, not to the behavior of \emph{vectors} in a representation.

\subsubsection{Poisson transform}\label{sssect-Pm} The Poisson
transform in this section provides us with $\E_s$ as a realization of
$\V{-\om}s$. It and its inverse can be described with the function
$R(t;z)^{1-s}$ in \eqref{R-def} as the kernel function. (For more
details see \S2.3 of~\cite{BLZ}.)\medskip

On $\V0s$ the Poisson transformation is the linear $G$-equivariant map
given in the line model by the simple formula
\be \poiss s \al(z) \= \frac 1\pi \int_{-\infty}^\infty \Bigl( \Bigl(
\frac{t-x}y\Bigr)^2 +1\Bigr)^{s-1} y^{-1+s} \al(t) \, dt \=
\int_{-\infty}^\infty R(t;z)^{1-s}\al(t)\, \frac{dt}\pi\,. \ee
The image is in $\E_s$, since $R(t;\dv)^{1-s}$ is in $\E_s$ for all
$t\in \RR$. Since $R(\dv;z)^{1-s}$ is an element of $\V\om s$ (line
model), the $G$-equivariance follows from
\be\label{Requiv}
R(\dv ; g z ) |_{2s}\, g \= R(\dv;z)^s\qquad\text{ for all }g\in
\Gm\,.
\ee
A comparison of this invariance property with \eqref{pq-Ginv} shows
that $R(\dv;\dv)^s$ is similar to $p_s$ and $q_s$.

We can write the Poisson transform as
\be \poiss s \al (z) = \langle R(\dv; z)^{1-s},\al \rangle\,.\ee
This can be used to define the
\indlbl{Po}{Poisson transformation}Poisson transformation as a linear
map
\indlbl{Potr}{$\poiss s$}$\poiss s :\V{-\om}s\lra\E_s$, satisfying
$\poiss s (\al\,|\,g) = (\poiss s \al)\,|\,g$ for all $g\in G$. The
following diagram, involving the isomorphism $I_s$ in \eqref{Idef},
commutes:
\be \label{P-I}
\begin{aligned}& \xymatrix@=.2cm{ \V{-\om}s \ar[dd]^{I_s} \ar[rrd]
^{\poiss s}
\\
&& \E_s = \E_{1-s}
\\
\V{-\om}{1-s} \ar[rru]^{\poiss{1-s}} }
\end{aligned}\ee
We also have
\begin{align}
\label{Poi-R} \bigl(\poiss s R(\dv;z')^s \bigr)(z) &\= p_s(z,z')\,;\\
\label{IR0}
I_s \, R(\dv;z')^s &\= R(\dv;z')^{1-s}\,.
\end{align}
(See (2.25) and (2.32) in \cite{BLZ}.)
In the other models, the Poisson kernel has the form:\indind{$R^\prj$,
$R^\ci$}{R-othermodels}
\beqabclbl{R-othermodels}
\begin{align}
\label{Rproj}
R^\prj(\z;z)^{1-s} &\= y^{s-1} \biggl( \frac{\z-i}{\z-z}\biggr)^{1-s}
\biggl( \frac{\z+i}{\z-\bar z} \biggr)^{1-s} \=
\biggl(\frac{R(\z;z)}{R(\z;i)} \biggr)^{1-s}\,,\\
\label{Rcirc}
R^\ci(\x;w)^{1-s} &\= \biggl( \frac{ 1-|w|^2}{(1-w/\x)(1-\bar w\x)}
\biggr)^{1-s}
\end{align}
\eeqabc


\sssect{Bijectivity}Crucial for these notes is that
$\poiss s: \V{-\om}s \lra \E_s$ is an isomorphism of $G$-modules.
This follows from the next result of Helgason (Theorem 4.3 in
\cite{Hel}) and the $G$-equivariance of~$\poiss s$.
\begin{thm}\label{thmHel} The Poisson transformation
$\poiss s:\V{}s \to \E_s$ is an isomorphism of $G$-modules for all
$s\in \CC$ with $0<\re s<1$.
\end{thm}
Thus, $\E_s$ is a model of the principal series representation. This
model has several advantages: the action of $G$ involves no
automorphy factor at all, the model does not give a preferential
treatment to any point, and all vectors correspond to actual
functions, with no need to work with distributions or hyperfunctions.


Theorem~3.2 in~\cite{BLZ} and the discussion preceding it give an
explicit way to describe the inverse of the Poisson transformation:
\begin{thm}\label{thmPoiss-inv}Let $u\in\E_s$, and $z_0\in\uhp$. Then
the hyperfunction $\al$ on $\p\uhp=\proj\RR$ represented by the
following function $g$ on $U\sm\proj\RR$ for a neighborhood $U$ of
$\proj\RR$ in~$\proj\CC$
\[ \renewcommand\arraystretch{1.4} g(\z)\= \left\{\begin{array}{rcll}
\int^\z_{z_0} \bigl[u,\, R^\prj(\z;\dv)^s\bigl] &+& u(z_0)\,
R^\prj(\z;z_0)^s\qquad&\text{if $\z\in\uhp\,$,}\\
\int_{\bar\z}^{z_0}\bigl[ R^\prj(\z;\dv)^s,\,u\bigr]&&
&\text{if $\z\in\lhp$.}\end{array}\right. \]
is independent of the choice of the base point $z_0$, and
$u=\poiss s\al\,$.
\end{thm}

\sssect{Polynomial growth} We define
\indlbl{E*}{$\E^{-\infty}_s$; $\E^\infty_s$;
$\E^\om_s$}$\E^{-\infty}_s$, $\E^\infty_s$ and $\E^\om_s$ as the
images under $\poiss s$ of $\V{-\infty}s$, $\V\infty s$ and
$\V\om s$, respectively. For $\E^{-\infty}_s$ we can indicate here an
independent characterization: We say that a function~$f$ on~$\disk$
has
\indlbl{pg}{polynomial growth}\emph{polynomial growth} if
$\left( 1-|w|^2\right)^a f(w) = \oh(1) $ as $|w|\uparrow 1$ for some
$a\in \RR$. For functions on~$\uhp$, this corresponds to
$z\mapsto \bigl( \frac y{|z+i|^2} \bigr)^a f(z)$ being bounded for
some~$a$.
\begin{thm}\label{thmLew} {\rm (Lewis; Theorem 4.1 and Theorem 5.3 in
\cite{Le})} \ Let $0<\re s<1$. The space
$\E_s^{-\infty}=\poiss s\bigl(\V{\infty}s\bigr)$ consists of the
functions in $\E_s$ having at most polynomial growth.
\end{thm}

\subsection{Boundary germs and transverse Poisson
transform}\label{sect-bgtPt}
In a comparison of the eigenfunctions $P_{\!s,n}$ and $Q_{s,n}$
introduced in \eqref{pqsndef}, a nice property of $P_{\!s,n}$ is that
it is defined on the whole of $\disk$, whereas $Q_{s,n}$ has a
singularity at~$0$. On the other hand, near the boundary $\p\disk$
the expression in \eqref{qsn-def} of $Q_{s,n}$ in terms of a
hypergeometric function implies a simple asymptotic relation
$Q_{s,n}(re^{i\th} ) \sim c \, (1-r^2)^s \, e^{in\th}$ as
$r\uparrow1$, whereas $ P_{\!s,n}(re^{i\th})$ has a more complicated
behavior at the boundary. We can observe a similar distinction
between the eigenfunctions $i_{s,\al}$ and $k_{s,\al}$, with
$\al\neq 0$, in \eqref{ikdef}. The asymptotic behavior of the
modified Bessel functions implies that $k_{s,\al}(z)$ is quickly
decreasing as $y\rightarrow\infty$, whereas $i_{s,\al}(z)$ has
exponential growth. Near $\RR\subset\p\uhp$ however we have
$i_{s,\al}(z) \sim e^{i\al x}\, y^s$ as $y\downarrow0$, whereas
$k_{s,\al}$ has a more complicated behavior.

We capture the special boundary behavior of $i_{s,\al}$ and $Q_{s,n}$
by defining in Subsection~\ref{ssect-bg} a space of eigenfunctions on
$\Om\cap \uhp$ for a neighborhood $\Om$ of $\p\H$ in $\proj\CC$ with
a special behavior near the boundary. Actually, we use germs of such
eigenfunctions by taking an inductive limit over all such
neighborhoods~$\Om$. In this way we define a space of boundary germs
$\W\om s$ isomorphic to $\V\om s$. The isomorphism
$\V\om s\lra \W\om s$ is described explicitly in
Subsection~\ref{ssect-tPm} by an operator that we call the
``transverse Poisson transformation''. In our study of cohomology
groups the space $\W\om s$ will turn out to be an excellent model
of~$\V\om s$.

For cohomology with coefficients in $\V\infty s$ we shall also need an
isomorphic space $\W\infty s$. This cannot be a space of germs of
eigenfunctions. In Subsection~\ref{ssect-bj} we will define it as a
space of expansions.

This whole section is a brief presentation of results discussed in
much more detail in \S4 and \S6 of~\cite{BLZ}.


\subsubsection{Boundary germs}\label{ssect-bg} We turn to
$\ld_s$-eigenfunctions only defined near the boundary $\p\H$ of the
hyperbolic plane. Our aim is to use such functions to define a space
$\W\om s$ isomorphic to $\V\om s$.


\sssect{The space of all boundary germs}
\indlbl{bg}{boundary germ}Put
\indind{$\F_s$}{F-def}
\be\label{F-def}
\F_s \= \limind\Om \E_s\left( \Om\cap\H \right)\,, \ee
where $\Om$ runs over the neighborhoods of $\p\H $ in $\proj\CC$. This
is a large space. The action of the group~$G$ is induced by
$f\,|\,g(z) = f\circ g(z) = f(gz)$ on representatives~$f$. We
identify $\E_s$ with its image in~$\F_s$.

Functions representing an element of $\F_s$ may grow fast near the
boundary. We define a subspace $\W\om s$ by prescribing the boundary
behavior:

\begin{defn}\label{Wom-def}The space
\indlbl{Wom}{$\W\om s$, $f^\ci$, $\W\om s(I)$, $f^\prj$,
$\tilde f$}$\W\om s$ is the subspace of $\F_s$ represented by
functions $f\in \E_s(\Om\cap \disk)$ for some neighborhood $\Om$ of
$\uc$ in $\proj\CC$ such that
$f(w)=2^{-2s}\allowbreak \bigl(1-\nobreak |w|^2\bigr)^s 
\allowbreak f^\ci(w)$, where $f^\ci \in C^\om(\Om)$.
\end{defn}

In the projective model there is a similar definition, with
$f(z)=\bigl( \frac y{|z+i|^2}\bigr)^s f^\prj(z)$ where $f^\prj$ is
real analytic on a neighborhood of $\proj\RR$ in $\proj\CC$. The
space $\W\om s$ is invariant under the action of $G$ in~$\F_s$.

The definition can be localized to define $\W\om s(I)$ for open sets
$I\subset\p\H$. Then $f^\ci$ or $f^\prj$ is real analytic on a
neighborhood $\Om$ of $I$ in $\proj\CC$. For $I\subset \RR$, the line
model is most convenient. Then each element of $\W\om s(I)$ is
represented by $f\in \E_s(\Om\cap\uhp)$ for a neighborhood $\Om$ of
$I$ in $\CC$, and $f(z) = y^s \tilde f(z)$ on $\Om \cap\uhp$ for some
real analytic function $\tilde f$ on~$\Om$.

We use the notation $\W\om s[F]=\W\om s(\H\sm F)$ for finite subsets
$F\subset\p\H$, and define\indind{$\W\om s[F]$; $\W\fs s$}{Wfs-def}
\be\label{Wfs-def}\W\fs s \= \limind F \W\om s(\p\H \sm F)\,, \ee
like in \eqref{V[]} and~\eqref{Vfs}. For a given $f\in \W\fs s$, the
set of
\indlbl{sing1}{singularity}singularities
\indlbl{bsing}{$\bsing$}$\bsing f$ is the minimal finite set
$F\subset\p\H$ such that $f\in \W\om s[F]$. The {\sf Bd} in this
notation reminds us that we consider singularities on $\p\H$, and not
at points of $\H$ near $\p\H$ where the functions $f^\ci$ or $f^\prj$
may have a singularity.


\sssect{Decomposition}Suppose $f\in \E_s(U)$, with
$U=\{w\in \disk\;:\; 1-\dt<|w|<1\}$, represents a germ in~$\F_s$.
Taking a closed curve $C$ in the annulus $U$ going round once in the
positive direction, with the Green's form in~\eqref{[]def} we form
the integral
\[ \int_C [ f, q_s(\dv,w)]\,.\]
This integral represents functions $u\in \E_s(I)$ and $v\in \E_s(E)$,
where $I$ is the region inside the curve $C$, and $E$ the annulus
outside~$C$. In Proposition~4.2, \cite{BLZ}, we show that $v$
represents an element of $\W\om s$, which vanishes if $f\in \E_s$.
Moving the curve $C$ closer and closer to $\uc=\p\disk$, we see that
$u$ extends to $\disk$ and is an element $u\in \E_s$.
Theorem~\ref{thm-psC} shows that $\frac1{\pi i}f=u-v$. Thus we have
obtained $\F_s = \E_s+\W\om s$. If the original function $f$ is in
$\E_s$, then $v=0$ and $u=\frac1{\pi i}f$, also by
Theorem~\ref{thm-psC}. So $\E_s \cap \W\om s=\{0\}$, and
\be\label{F-decomp} \F_s \= \E_s\oplus \W\om s\,.\ee

\sssect{Restriction to the boundary}\indlbl{resbd}{restriction to
boundary}If $w$ represents an element of $\W\om s$, then
$f(w) = (1-\nobreak |w|^2)^s f^\ci(w)$ near the boundary, with
$f^\ci$ extending analytically across the boundary. Thus
\indlbl{rho-def}{$\rho_s$}$\rho_s f (\x)= f^\ci(\x)$ is a well defined
analytic function on $\uc$, which is an element of the circle model
of $\V\om s$. This restriction map $\rho_s$ intertwines the actions
of $G$ in $\W\om s$ and $\V\om s$. We should note that $f^\ci$ is
real analytic on a neighborhood $\Om$ of $\uc$ in $\proj\CC$, and
that $\ph = \rho_s f$ is a real analytic function on $\uc$ extending
as a holomorphic function on some neighborhood $\Om_1$ of $\uc$ in
$\proj\CC$. These functions $\ph $ and $\tilde f$ coincide on $\uc$,
not on the whole intersection~$\Om_1\cap \Om$.

In the upper half plane model of $\H$, we obtain $\rho_s f = f^\prj$
in the projective model of $\V\om s$ on $\proj\RR$. The restriction
$\rho_s$ also gives linear maps $\rho_s : \W\om s(I) \lra \V\om s(I)$
for open $I \subset \p\H$. In particular, for $I\subset\RR$, we
obtain $\rho_s f = \tilde f$ in the line model.

\subsubsection{Transverse Poisson map}\label{ssect-tPm}
The restriction map $\rho_s:\W\om s\lra\V\om s$ is bijective. In \S4.2
of~\cite{BLZ} we explicitly describe the inverse, in two different
ways.

One way is by an integral transform, with the following expression
\indind{$\tpoiss s$}{Wdef}\indind{$\bb(s)$}{btd} in the three models:
\beqabclbl{Wdef}
\begin{align}
\label{Wdef-ln}
\bigl(\tpoiss s\ph\bigr)(z) &\= \frac1{i\,\bb(s)} \int_{\bar z}^z
    R(\z;z)^{1-s}\,\ph(\z)\, d\z\,,
    \displaybreak[0]\\
\label{Wdef-prj}
\bigl(\tpoiss s \ph^\prj\bigr)(z) &\= \frac1{i\,\bb(s)}\,\int_{\bar
z}^z R^\prj(\z;z)^{1-s}\ph^\prj(\z)\, \frac{d\z}{1+\z^2}\,,
  \displaybreak[0]
\\
\label{Wdef-ci}
  \bigl(\tpoiss s \ph^\ci\bigr)(w) &\= \frac1{2\,\bb(s)}
  \int_w^{1/\bar w} R^\ci(\eta;w)^{1-s}\ph^\ci(\eta)\,
  \frac{d\eta}\eta\,,\\
\label{btd}
\text{where }\qquad \bb(s)&\= \Beta\Bigl(s,\frac12\Bigr)\=
\frac{\sqrt\pi\,\Gf(s)}{\Gf(s+\frac12)}\,.
\end{align}
\eeqabc
An element $\ph\in \V\om s(I)$ for some open $I\subset\p\H$ extends
holomorphically to some neighborhood $\Om$ of $I$ in $\proj\CC$. The
integrals in \eqref{Wdef} define $\tpoiss s \ph $ on
$\Om \cap\bar\Om\cap\H$, representing a germ that can be shown to be
an element of $\W\om s(I)$. (By $\bar\Om$ we denote the image under
complex conjugation.) On the other hand, Theorem~4.7 in~\cite{BLZ}
gives also an integral representation of $\ph$ in terms
of~$u=\tpoiss s \ph$, showing that if
(in the line model) $u=y^s A$ with $A$ real analytic on a simply
connected open set $\Om=\bar\Om$ intersecting~$\RR$ then $\ph$ is
holomorphic on~$\Om$.

The integral transformation in~\eqref{Wdef} has the same kernel
function as that in the Poisson transformation. The path of
integration is different. We call $\tpoiss s$ the
\indlbl{tP}{transverse Poisson transformation}\emph{transverse Poisson
transformation}.
\begin{thm}The restriction map $\rho_s:\W\om s(I)\lra \V\om s(I)$ is
an isomorphism for each open set $I\subset\p\H$. Its inverse is given
by~$\tpoiss s$.
\end{thm}

The other way to describe the transverse Poisson transformation works
locally with the line model. The action of $G$ allows restriction to
an interval $I\subset\RR$. Any $f\in \W\om s(I)$ is of the form
$f(z) = y^s A(z)$, with $A$ real-analytic on a neighborhood $\Om$ of
$I$ in $\RR$. Let $\ph=\rho_s f$. Then it turns out that the fact
that $f\in \E_s(\Om\cap\uhp)$ implies that $A$ has the expansion
\be\label{omw-A-exp} A(z) \= \sum_{k=0}^\infty \frac{(-1/4)^k\,
\Gf\bigl(s+\frac12\bigr)}{k!\, \Gf\bigl(s+\frac12+k\bigr)}\,
\ph^{(2k)}( x)\, y^{2k}\,.\ee
(See Theorem~4.6 in~\cite{BLZ}.) If $\Om_x\subset \Om$ is an open
neighborhood of $x\in I$ on which the power series of $\ph$ at $x$
converges, then \eqref{omw-A-exp} converges on $\Om_x$ as well.

This relation between the expansions of $A$ and $\ph$ illustrates that
being a $\ld_s$-eigen\-func\-tion of $\Dt$ is a very strong property.
Note that the description in \eqref{Wdef} shows that if $\ph$ is
holomorphic on $\Om$, then $A$ is real-analytic on~$\Om$, but that
conversely if $A$ is real-analytic on $\Om_1$, we know only that
$\ph$ is holomorphic on some neighborhood $\Om$ of $I$ that may be
much smaller than $\Om_1$.

\sssect{Examples} For the functions in \eqref{ikdef}, \eqref{psn-def},
\eqref{R-def} and \eqref{pqs-def} we have:
\beqabclbl{ex1}
\begin{align}
\label{ex1a}
 \tpoiss s e^{i\al x}&\=i_{s,\al}&& \text{ on $\RR$ in the line
 model}\,,\\
\label{ex1b}
  \tpoiss s \eph_{s,n}&\=\frac{(-1)^n \, \Gf(s+\frac12)}{\sqrt \pi\,
  \Gf(s+n)} \, Q_{s,n}&&\text{ on $\p\H$}\,,\\
\label{ex1c}
   \tpoiss s R(\dv;z)^s\,(z')&\=\bb(s)^{-1}\, q_s(z,z')
   &\quad& \text{ on $\RR$ in the line model}\,.
\end{align}
\eeqabc
The first two examples are easily checked by computing the restriction
$\rho_s$ of the left hand side. The third example is equation~(4.19)
in~\cite{BLZ}. See \S A.3 in the appendix of~\cite{BLZ} for more
examples.



\sssect{Splitting of $\E_s^\om$}For $s\neq\frac12$, Proposition~6.3
in~\cite{BLZ} gives the following description of
$\E^\om_s=\poiss s\V\om s$:
\be\label{splitting}
\begin{aligned}
\poiss s \ph &\= c(s)\, \tpoiss s \ph
+c(1-s)\,\tpoiss {1-s} I_s \ph\,, \\
c(s)&\= \frac{\tan\pi s}{\pi}\, \bb(s)\,,
\end{aligned}
\ee\indind{$c(s)$}{splitting}
with the intertwining operator $I_s$ in \eqref{Idef}, and $\bb(s)$ as
in~\eqref{btd}. This implies that each of the isomorphic $G$-spaces
$\E^\om_s$, $\W\om s$ and $\W\om{1-s}$ is contained in the sum of the
other two, and that each two of these spaces have
intersection~$\{0\}$.


\sssect{Duality} The $G$-invariant duality of
$\V\om s \times\V{-\om}{1-s}\lra\CC$ in \eqref{dual-ci} can be
transported to a $G$-invariant duality
$ \W\om s \times \E_{1-s}\lra \CC$ by\indind{duality;
$\langle \dv,\dv\rangle$}{}
\be\label{WEdual}
\langle \tpoiss s \ph ,\poiss {1-s} \al \rangle \= \langle \ph,\al
\rangle
\qquad(\al\in \V{-\om}{1-s}\,,\ph\in \V\om s)\,. \ee

In Proposition~4.8 in~\cite{BLZ} it is shown that for $u\in \E_{1-s}$
and $f\in \W\om s$ we can describe this duality with the Green's form
of \S\ref{sssect-Grf}:
\be \label{WPdual}
\langle f,u \rangle \= \frac{\bb(s)}{\pi i} \int_C [f,u] \= -
\frac{\bb(s)}{\pi i} \int_C [u,f] \,, \ee
where $C$ is homotopic to $\p\H$ in the domain of a representative of
$f$, going around once in the positive direction. Actually,
$\int_C [f,u]$ is well defined for $f\in \F_s$, $u\in \E_{1-s}$,
being zero for $f$ in the component $\E_s=\E_{1-s}$ of the splitting
$\F_{1-s} = \E_{1-s} \oplus \W\om{1-s}$ in \eqref{F-decomp}. The
second equality in \eqref{WPdual} follows from \eqref{som[]} and the
fact that $C$ is a closed curve.


\subsubsection{Boundary jets}\label{ssect-bj}
In Chapter~\ref{ch-V} we study cohomology with differentiable
coefficients. We need a substitute $\W p s$ for $p=2,3,\ldots,\infty$
for the space $\W\om s$ of boundary germs. In \S4.4 of \cite{BLZ} we
have constructed these spaces as a quotient. We recall the
definitions and main results. For $\allp$ we consider the space of
functions $f\in C^2(\disk)$ for which
$\tilde f(w)=(1-\nobreak|w|^2)^{-s}f(w)$ extends to a $C^p$ function
on some neighborhood of $\uc$ in $\CC$. By $\tilde\Dt_s$ we denote
the differential operator on~$\tilde f$ corresponding to $\Dt-\ld_s$
on~$f$. We define
\indlbl{Gd}{$\G^p_s$}$\G^p_s$ and
\indlbl{Nd}{ $\N^p_s$}$\N^p_s$ as the subspaces of functions
$f=(1-\nobreak |w|^2)^s\tilde f$ that satisfy the conditions
\beqabclbl{Dts-s-est}
\be \label{Dts-s-est-a}
\tilde\Dt_s \tilde f (w) \= \begin{cases}
\o\bigl( (1-|w|^2)^p\bigr)&\text{ if }p\in \NN\,,\\
\o\bigl( (1-|w|^2)^q\bigr)&\text{ for all }q\in \NN\text{ if
}p=\infty\,,\\
0&\text{ if }p=\om\,,
\end{cases}
\ee
respectively
\be \tilde f (w) \= \begin{cases}
\o\bigl( (1-|w|^2)^p\bigr)&\text{ if }p\in \NN\,,\\
\o\bigl( (1-|w|^2)^q\bigr)&\text{ for all }q\in \NN\text{ if
}p=\infty\,,\\
0&\text{ if }p=\om\,,\end{cases}
\ee\eeqabc
as $|w|\rightarrow 1$ in some annulus $1-\e<|w|<1$. In~\cite{BLZ},
Lemma~4.10 it is shown that $\N^p_s\subset\G_s^p$. We define the
space of \emph{boundary jets}
\indlbl{Wpdef}{$\W p s$}$\W p s$ as the quotient $\G^p_s/\N^p_s$ for
$p=2,3,\ldots,\infty$, so that the following sequence is exact by
definition:
\be\label{NGWex}
0\lra \N^p_s\lra\G^p_s\lra\W p s\lra0\;.\ee
For $p=\om$ this agrees with our previous definition of $\W\om s$
because a function $f\in \G^\om_s$ is in $\N^\om =  C^2_c(\disk)$ if
and only if it represents the zero element of $\W\om s$. The group
$G$ acts on $\G^p_s$ and $\N^p_s$ by $(f\,|\,g)(w) = f(gw)$. This
induces an action in $\W p s$.

These definitions also work locally.
\indlbl{GNI}{$\G^p_s(I)$, $\N^p_s(I)$}For $I\subset\uc$ open we define
$\G^p_s(I)$, $\N^p_s(I)$ as above (with $f$ still defined on all
of~$\disk$) but with the extendability across $\uc$ and the growth
conditions~\eqref{Dts-s-est} near~$\uc$ required only near~$I$. Thus,
$\G^p_s$ and $\N^p_s$ are sheaves on~$\uc$. We define $\W p s$ as the
quotient sheaf. One can show that $\W p s(I) = \G^p_s(I)/\N^p_s(I)$
for all $I$. In the upper half plane model we have corresponding
definitions with the factor $(1-\nobreak|w|^2)^s$ replaced by
$\bigl(\frac y{|z+i|^2}\bigr)^s$
(or simply by $y^s$ on $I\subseteq\proj\RR$ with $\infty\notin I$).

We define $G$-equi\-variant sheaf morphisms
\indlbl{r2}{$\rho_s$}$\rho_s:\G^p_s\lra \V p s$ by sending
$f\in\G^p_s(I)$ to the restriction to $I$ of a $C^p$ extension of
$2^{2s}(1-|w|^2)^{-s}f(w)$ (resp.~of
$\bigl(y/|z+i|^2\bigr)^{-s}f(z)$). In Theorem~4.11 of~\cite{BLZ} we
prove:
\begin{thm}The restriction $\rho_s$ induces a sheaf isomorphism
$\rho_s:\W p s\lra \V p s$ for $\allp$.
\end{thm}

Notice that we have global representatives in
$\G^p_s(I) \subset C^2(\H)$ of elements of $\W p s(I)$, even if
$I\subset \p\H$ is a tiny interval. We impose twice differentiability
in all of $\H$ in order to be able to apply $\Dt$ freely. Even in the
analytic case $p=\om$ the representatives $f\in\G^\om_s(I)$ need not
be analytic on all of~$\H$, and satisfy $\Dt f=\ld_sf$ only near the
boundary.

\begin{defn}\label{singdef}
For any $f:\H\lra\CC$ the set of \emph{singularities} $\sing f$ of~$f$
is the complement of the maximal open set $U\subset \H$ such that
$f\in \E_s(U)$.
\end{defn}

This is a rather broad notion of singularity. It depends on the
spectral parameter $s$, and even an analytic function may have
singularities in our sense. For $f\in \G^\om_s$ the set $\sing f$ is
a compact subset of~$\H$. This set may be empty. The function
$i_{s,0}(z) = y^s$ is an element of $\E_s$ and of $\G^\om_s(\RR)$,
with $\sing i_{s,0}=\emptyset$. Note that $ i_{s,0}$ represents an
element of $ \W\om s(\RR) $ which can be considered as an element of
$\W\fs s$, as defined in \eqref{Wfs-def}. As such $\bsing i_{s,0}
= \{\infty\}$.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Averages}\label{ssect-av}
For $v$ in any $G$-module $V$ all finite sums
$v\,|\sum_i g_i = \sum_i v\,|\,g_i$ converge. Some infinite sums
converge as well, for certain modules. In this section we discuss
infinite sums that will be used in the next chapters. It has turned
out that these operators form a useful tool when dealing with
transfer operators.
(See~\cite{BM}, \cite{LZ}.)

The infinite sums that we discuss in this section are\indind{ $\av g$,
$\av g^+$, $\av g^-$}{}
\be\label{av-def}
 \av g^+ \= \sum_{n=0}^\infty g^n\,,\quad \av g^- \=
-\sum_{n=-\infty}^{-1} g^n\,,\quad \av g \= \sum_{n=-\infty}^\infty
g^n\= \av g^+-\av g^-\,, \ee
for $g\in G$ with infinite order. If we can make sense of the
convergence of such sums, the \indlbl{osa}{one-sided
average}\emph{one-sided averages} $\av g^+$ and $\av g^-$ provide us
with a substitute for $(1-\nobreak g)^{-1}$. The
\indlbl{av}{average}\emph{average} $\av g$ produces a $g$-invariant
vector.

The elements of $G$ of infinite order are either hyperbolic or
parabolic. We treat these two cases separately, and consider the
one-sided averages and the spaces of invariants for spaces in
$\V{}s$, in particular for $\V\om s$.

\sssect{Notation} We shall use both $f\,|\,\av g$ and $\av g(f)$ to
denote the average of~$f$ over the powers of~$g$. The latter notation
emphasizes the average as an operator, whereas the former stresses
that $\av g$ is an element of the completion of the group ring
of~$\Gm$.

\subsubsection{Invariants and averages for hyperbolic
elements}\label{ssect-avhyp} We start with the easiest case, where
$g$ is hyperbolic.\indind{hyperbolic element}{}

Any hyperbolic $\eta\in G$ leaves fixed two points of $\proj\CC$,
which are situated on~$\proj\RR$: The
\indlbl{hyp}{hyperbolic fixed point}\indlbl{rfp}{repelling fixed
point; $\al(\eta)$}\emph{repelling fixed point} $\al(\eta)$ and the
\indlbl{afp}{attracting fixed point; $\om(\eta)$}\emph{attracting
fixed point} $\om(\eta)$. The latter is characterized by
$\lim_{n\rightarrow\infty} \eta^n x = \om(\eta)$ for all
$x\in \proj\RR\sm\{\al(\eta)\}$. By conjugation in~$G$, we can
arrange $\eta=\matc{\sqrt t}00{1/\sqrt t}$ with $t>1$. Then
$\al(\eta)=0$, $\om(\eta)=\infty$.\smallskip


By \indlbl{inv}{invariants;$V^g$}$V^g$ we denote the elements of $V$
invariant under $g\in G$ for any $G$-module~$V$.
\begin{prop}\label{prop-hyp-inv}The spaces $(\V p s)^\eta$ are $\{0\}$
for $\allp$, and also the spaces $(\V\om s[0])^\eta$ and
$(\V\om s[\infty])^\eta$ are zero.
\end{prop}
See \eqref{V[]} for the definition of $\V\om s[\x]$.
\begin{proof}It suffices to consider $\V0s(\RR)=\V0s[\infty]$ and
$\V0s(\proj\RR\sm\{0\})=\V0s[0]$. Let $f\in \V0s(\RR)$ be given in
the line model. Then $t^s f(tx)=f(x)$ for all $x\in \RR$. There is a
periodic function $p$ on~$\RR$ with period $\log t$ such that
$f(x) = x^{-s} p(\log x)$ for $x\in (0,\infty)$. This implies that
$\lim_{u\rightarrow-\infty}
p(u) = \lim_{u\rightarrow-\infty} e^{su} f(e^u) = 0$. Hence $f=0$ on
$(0,\infty)$, and analogously on $(-\infty,0)$. Conjugate with
$\matr0{-1}10$ to obtain the statement for $\V0s(\proj\RR\sm\{0\})$.
\end{proof}
With the isomorphism $\tpoiss s$, we obtain also that the
corresponding spaces $(\W p s)^\eta$ all vanish.


For larger spaces in the principal series, the spaces of
$\eta$-invariants are large. They contain the spaces of functions
transforming according to the character
$\matc{\sqrt y}00{1/\sqrt y} \mapsto y^{i\al}$ for all
$\al \in \frac{2\pi}{\log t}\ZZ$. For each $\al\in \RR$ we give in
(A.20) of~\cite{BLZ} functions $f^R_{s,\al}$ and $f^L_{s,\al}$ in
$\E_s$ that form a basis for the invariant functions for the
character specified by~$\al$. We have for $z=\rho e^{i\phi}\in \uhp$,
$\rho>0$, $0<\phi<2\pi$:\indind{$f^R_{s,\al}$}{frt}
\be\label{frt}
\begin{aligned}
f^R_{s,\al}(\rho e^{i\phi}) &\= \frac{\sqrt\pi\, \Gf(s+\frac12)}
{\Gf(\frac{s+i\al+1}2)\, \Gf(\frac{s-i\al+1}2)}\, \rho^{i\al}\,
(\sin\phi)^s\, \hypg21{\Bigl(\frac{s+i\al}2,\frac{s-i\al}2;\frac12;
 (\cos\phi)^2 \Bigr)}\\
&\qquad\hbox{}
- \frac{2\sqrt\pi\, \Gf(s+\frac12)}{\Gf(\frac{s+i\al}2)\,
\Gf(\frac{s-i\al}2)}\,\rho^{i\al}\, \cos\phi \, (\sin \phi)^s\\
&\qquad\qquad\hbox{} \cdot
\hypg21{\Bigl(\frac{s+i\al+1}2,\frac{s-i\al+1}2;\frac 32;
(\cos\phi)^2\Bigr)}\,,
\end{aligned}
\ee
and we obtain $f^L_{s,\al}$ by taking the sum of the two terms instead
of the difference. We have chosen the basis such that $f^R_{s,\al} $
represents an element of $\W\om s(0,\infty)$ and $f^L_{s,\al}$ an
element of $\W\om s(-\infty,0)$. We have
$\bsing f^R_{s,\al}=\{\infty\}\cup(-\infty,0]$, and $\bsing
f^L_{s,\al}=[0,\infty)\cup\{\infty\}$. Note that the first term in
\eqref{frt} is invariant under $z\mapsto-\bar z$, and the second term
anti-invariant.
\medskip


Let $\ph \in \V\om s$. For large $|x|$ it is of the form
$\ph(x) = |x|^{-2s} \ph_\infty(1/x)$ with $\ph_\infty$ the
real-analytic function on a neighborhood of~$0$ given in~\eqref{asy}.
For $x\neq0$ we have
\[\ph|_{2s}\,\eta^n (x) = t^{ns} \ph(t^n x) = t^{-ns} |x|^{-2s}
\ph_\infty(t^{-n}x^{-1})
\] as $n\rightarrow\infty$. Since $\re s>0$ and $t>1$, the series
$\sum_{n=0}^\infty t^{ns} \ph(t^n x)$ defining $ \av \eta^+ (\ph)(x)$
converges with exponential rapidity for $x\neq 0$, so that this
function is defined and real-analytic on $\proj\RR\sm\{0\}$. It may
have a singularity at~$0$, so in general it will belong to the larger
space $\V\om s[0]$. In fact, we may allow $\ph$ itself to belong to
this larger space, since then the convergence goes through. For
$\av \eta^-(\ph)$ we proceed similarly. Now the point $\infty$ may be
a singularity. We have obtained the following left inverses of
$1-\nobreak \eta$:
\be \label{av-eta} \av \eta^+ : \V\om s[0]\lra \V\om s[0]
\quad\text{ and }\quad \av \eta^- : \V\om s[\infty]\lra \V\om
s[\infty]\,. \ee
If $\ph \in \V\om s$, then the total average $\av \eta(\ph) =
\av \eta^+(\ph) - \av \eta^-(\ph)$ is defined and belongs to
$\V\om s[0,\infty]^\eta$.


\begin{prop}\label{prop-av}
The following three statements are equivalent for $\ph\in \V\om s$:
\be 1:\hbox{\ }\av\eta ^+(\ph) \in \V\om s\,,\qquad2:\hbox{\ }
\av\eta^-(\ph) \in \V\om s\,,\qquad3:\hbox{\ } \av \eta(\ph)\=0\,.\ee
\end{prop}
\begin{proof}(1)$\;\Rightarrow\;$(3): If $\av\eta^+ (\ph)\in \V\om s$,
then $\av \eta (\ph) = \av\eta^+(\ph) - \av\eta^-(\ph) 
\in \V\om s[\infty]^\eta=\{0\}$, by Proposition~\ref{prop-hyp-inv}.
(2)$\;\Rightarrow\;$(3) goes similarly.
(3)$\;\Rightarrow\;$(1)\&(2): If $\av \eta(\ph)=0$, then
$\av \eta^+(\ph) 
= \av \eta^-(\ph)  \in \V\om s[0]\cap\V\om s[\infty]=\V\om s$.
\end{proof}
\begin{cor}\label{cor-avh}The kernel of
$\av \eta: \V\om s \lra \V\om s[0,\infty]^\eta$ is equal to
$\V\om s\,|\,(1-\nobreak\eta)$.
\end{cor}
\begin{proof}
Clearly, $\V\om s\,|\,(1-\nobreak\eta)\subset \ker \av \eta$. If
$\av \eta(\ph)=0$, then
$\ph = \ph\,|\,\av \eta^+|(1-\eta) \in \V\om s\,|\,(1-\nobreak\eta)$.
\end{proof}
For other hyperbolic elements we have
\be \label{aveta-c}
 \av{g\eta g^{-1}}^+ : f \mapsto f\,|\,g\,\av \eta^+ \,g\,,\qquad \av
 {\eta^{-1}}^+ \= - \eta \,\av \eta^- \= - \av\eta^- \,\eta\,.
\ee


\sssect{Averages in $\G^p_s$} With the transverse Poisson
transformation these statements can be transformed into analogous
statements for $\G^ p _s$.

For $f\in \G^p_s$, with $\allp$, we use that, in the line model,
$f(z) = y^s \allowbreak |z|^{-2s} \allowbreak f_\infty(1/z)$ on
$\Om \cap\uhp$ where $f_\infty \in C^p(\Om)$ for some neighborhood
$\Om$ of $0$ in~$\CC$. This implies that
$\av \eta^+ f(z) = y^s \allowbreak |z|^{-2s} 
\sum_{n=0}^\infty t^{-ns} f_\infty\bigl( 1/t^nz\bigr)$ converges
absolutely, uniformly on compact sets in~$\uhp$. We see that for each
neighborhood $\Om_1$ of $\infty$ in $\proj\CC$ not containing $0$
there is $A\in \NN$ such that for $z\in C$ the function
\[ z \mapsto t^{-ns} f_\infty \bigl( 1/t^n z\bigr)\]
is in $C^p(\Om_1)$. This shows that
$\av \eta^+ f \in \G^p_s\bigl(\proj\RR\sm \{0\} \bigr)$. Similarly we
get $\av \eta^- f \in \G^p_s(\RR)$. Observe that $\av \eta^+ f$ is in
general not an element of $\G^p_s = \G^p_s(\proj\RR)$.
\smallskip

The following lemma will be needed for the proof of Theorem~\ref{thmA}
(see Lemma~\ref{lem-ch-reg}).
\begin{lem}\label{lem-hyp-cp}Suppose that for $f\in \G^\om_s$ the set
$\sing \av \eta^+ f $ is compact in $\uhp$. Then
$\av \eta^+ f \in \nobreak \G^\om_s$.
\end{lem}
\begin{proof}
The singularities of $\av \eta^- f$
(see Definition~\ref{singdef}) are contained in
\[ \bigcup_{n\leq -1} \eta^{-n} \sing f\,.\]
Since $\sing f $ is compact there exists $\e>0$ such that
\[ \sing \av \eta^- f \quad\subset\quad S_\e \,:=\, \bigl\{ z\in
\uhp\;:\; y>\e\,,\; \e<\arg z<\pi-\e\bigr\}\,.\]
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Decreasing $\e>0$ if necessary, we arrange that the compact set
$\sing \av \eta^+ f$ is also contained in $S_\e$. So the
$\eta$-invariant function $\av \eta f = \av \eta^+ f - \av \eta^- f$
determines an element $h\in \E_s(\uhp\sm S_\e)$, which satisfies
$h(tz) = h(z)$ whenever $z, tz\not\in S_\e$. Hence $h$ extends as an
element of $\E_s(\uhp)^\eta$. Thus, $h$ has a Fourier expansion with
$\al$ running through $\frac{2\pi}{\log t}\ZZ$:
\be h(z) \= \sum_\al h_\al(z)\,,\qquad h_\al(z) \= \frac1{\log t}
\int_0^{\log t} e^{-i\al u} h(e^uz)\, du\,. \ee
For each such~$\al$, the function $h_\al$ is a linear combination of
the functions $f^R_{s,\al}$ and $f^L_{s,\al}$ in~\eqref{frt}.

The function $h$ represents an element of $\W\om s[0,\infty]$. Hence
near $\RR$ it has the form $h(z) = y^s \tilde h(z)$, with $\tilde h$
real analytic on a neighborhood of $\RR\sm\{0\}$ in $\CC$. For $x>0$
and small values of $\th>0$:
\begin{align*}
h_\al(xe^{i\th}) &\= (x\sin \th)^s \frac1{\log t} \int_0^{\log t}
e^{-i\al u} \tilde h(e^u x e^{i\th})\, du\\
&\= (x\sin\th)^s \cdot\left(\text{real-analytic on a neighborhood of
$(0,\infty)$}\right)\,.
\end{align*}
So $h_\al$ represents an element of $\W\om s(0,\infty)$, and the
coefficient of $f^L_{s,\al}$ is zero. Proceeding similarly near
$(-\infty,0)$ we obtain that the coefficient of $f^R_{s,\al}$
vanishes as well. This works for all
$\al \in \frac{2\pi}{\log t}\ZZ$, and hence $h=0$. This means that
$\av \eta f \in \G^\om_s(\RR)$. Since $\av \eta^- f \in \G^\om(\RR)$,
we have also $\av \eta^+  \in \G^\om_s(\RR)$. For all
$f\in \G^\om_s$, we have $\av \eta^+ f \in
\G^\om_s\bigl( \proj\RR\sm\{0\}\bigr)$. Hence $\av \eta^+ f \in
\G^\om_s(\proj\RR)=\G^\om_s$.
\end{proof}


\subsubsection{Invariants and averages for parabolic
elements}\label{ssect-invavpar}
For parabolic elements of $G$ the invariants in subspaces of $\V{}s$
with high regularity vanish, like for hyperbolic elements. The
convergence of the averages is more delicate than in the hyperbolic
case.
\smallskip


Each parabolic element of $G$ is conjugate in $\PSL_2(\RR)$ to
\indlbl{T}{parabolic element; $T$}$T=\matc1101$ or to $T^{-1}$.
Parabolic elements have only one
\indlbl{pfp}{parabolic fixed point}fixed point, situated on
$\proj\RR$. The element $T$ fixes $\infty\in \proj\RR$.
\begin{prop}\label{prop-Tinv0}The space $(\V p s)^T$ is zero for
$\allp$.
\end{prop}
\begin{proof}Each $\ph \in \V p s$ (line model) satisfies
$\ph(x) = x^{-2s} \ph_{\infty}(1/x) = \o(1)$ as $x\rightarrow\infty$. If $\ph$
is also periodic, then it must vanish identically.
\end{proof}

For $\ph\in \V p s$ (line model with $\allp$), the averages
\be \av T^+(\ph) (x) \= \sum_{n=0}^\infty \ph(x+n)\,,\qquad \av T^-
(\ph)(x) \= - \sum_{n=-\infty}^{-1} \ph(x+n)\,, \ee
converge if $\re s>\frac12$ or if $\re s>0$ and the number
$C=\ph^\prj(\infty)$ vanishes. In general we have
$\ph(x) =C|x|^{-2s}+ \oh (|x|^{-2s-1})$ as $|x|\rightarrow\infty$ and
we define
(for $\re s>0$, $s\neq\frac12$)
\be \label{reg-av}\begin{aligned} \av T^+(\ph)(x) &\=
\sum_{n=0}^\infty \Bigl(\ph(x+n)
  \,-\,\frac C{(n+1)^{2s}}\Bigr)\;+\;C\,\z(2s)\;, \\
\av T^-(\ph)(x) &\= -\,\sum_{n=1}^\infty \Bigl(\ph(x-n)
  \,-\,\frac C{n^{2s}}\Bigr)\;-\;C\,\z(2s)\;.
  \end{aligned} \ee
Since differentiation only improves the convergence, we see that if
$\ph\in \V p s$, then $\av T^\pm \ph$ is in $C^p(\RR)$. So we have
$\av T^\pm : \V p s \rightarrow \V p s[\infty]=\V p s(\RR)$, and more
generally
\be\label{avrel} \av T^+\,:\,\V p s((b,a)_c)\,\rightarrow\,\V p
s(b,\infty),\quad \av T^-\,:\,\V p s((b,a)_c)\,\rightarrow \,\V p
s(-\infty,a+1) \ee
for $a,\,b\in\RR$ with $a<b$, where we use the convenient notation
$(b,a)_c$ for the ``cyclic interval" \indlbl{cint}{cyclic interval,
$(b,a)_c$}$(b,\infty) \cup \{\infty\} \cup (-\infty,a)\subset\proj\RR$.
 It is clear that these one-sided averages satisfy
\be\label{rel-avT} \ph\,|\,(1-T)\,|\,\av T^\pm \= \ph\,|\, \av T^\pm
\,|\,(1-T) \= \ph\,. \ee
Furthermore, if we denote by $C_m$
($0\leq m\le p)$ the coefficient of $x^m$ in the Taylor expansion of
$\ph|_{2s}\,\matr0{-1}10(x)$, then using the Euler-Maclaurin
summation formula or arguing as in \cite{LZ}, Chap.~III,
\S3, we find that the functions $\av T^+(\ph)$ and $\av T^-(\ph)$
have the one-sided asymptotic behavior
\be\label{asympt}
 \av T^\pm (\ph)(x) \= |x|^{-2s}\,\sum_{m=-1}^{p-1} C_m^\ast \,x^{-m}
 \;+\;\oh\bigl(|x|^{-2s-p}\bigr)
  \;\quad\text{ as $\;\pm x\rightarrow\infty$}
  \ee
(in the line model) with the coefficients $C_m^\ast$ in both cases
given explicitly by
\be\label{Cmast}
C_m^\ast \= \frac{(-1)^{m+1}}{m+2s} \sum_{k=0}^{m+1} B_k \, C_{m+1-k}
\, \binom{m+2s}k\,, \ee
where $B_k$ is the $k$th Bernoulli number. If $p=\infty$ or $p=\om$,
then \eqref{asympt} must be interpreted as an infinite asymptotic
expansion (not necessarily convergent for any $x$, even in the
analytic case).

For other parabolic $\pi\in G$, we define $\av \pi^\pm$ by conjugation
and the relations $\av {\pi^{-1}}^+ = - \pi \av \pi^-$ and
$\av {\pi^{-1}}^- = - \pi^{-1} \av \pi^+ = -\av\pi^+\pi$.

Using the transverse Poisson transformation~$\tpoiss s$
in~\S\ref{ssect-tPm} we can transport the one-sided averages to
$\W p s=\G^p_s/\N^p_s$ for $p=2,\ldots,\om$. But we can also define
the maps $\av T^\pm$ directly at the level of $\G^p_s$ in the obvious
way (for instance, $\av T^+f(z)$ is defined as
$\sum_{n=0}^\infty f(n+z)$ if $C=f^\prj(\infty)$ vanishes and
otherwise as
$\sum_{n=0}^\infty\bigl(f(n+z)-Cy^s/(n+\nobreak1)^{2s}\bigr)
 \,+\,C\z(2s)y^s$). We thus obtain maps as in \eqref{avrel} with
 $\V {}{}$ replaced by $\W {}{}$ or $\G$, still satisfying the
relations~\eqref{rel-avT}. The new aspect is that, as partners of the
asymptotic relations \eqref{asympt} on~$\RR$, we get new asymptotic
relations for $\av T^\pm f(x+iy)$ as $y\to\infty$.

\begin{lem}\label{lem-avWas}
Let $f\in \G^p_s(I)$, with $\allp$, for some interval
$I\subset\proj\RR$ containing~$\infty$. For $s\neq \frac12$, we have
\[ \av T^\pm f(z) \= \pm\,\frac{\sqrt
\pi\,\Gf(s-\frac12)}{2\,\Gf(s)}\,f^\prj(\infty)\,y^{1-s} \;+\;
\oh\bigl(f^\prj(\infty)\,x\,y^{-s}\bigr)\;+\;
\oh\bigl(y^{-s}\bigr)\;. \]
as $y\to\infty$, uniformly in $x$.
\end{lem}
\begin{proof} If $f^\prj(\infty)=0$, then $f(z)=\oh(y^s|z|^{-2s-1})$
for $|z|$ large, so
\[ \av
T^+f(z)\=\sum_{n=0}^\infty\oh\biggl(\frac{y^s}{|n+z|^{2s+1}}\biggr)
  \=\oh\biggl(y^s\int_{-\infty}^\infty\frac{dt}{(t^2+y^2)^{s+1/2}}\biggr)
  \=\oh\bigl(y^{-s}\bigr)\,. \]
To treat the general case, it suffices to consider one function $f$
with $f^\prj(\infty)\ne0$. We choose $f=F_s$, where
$F_s(z)=y^s|z|^{-2s}$. If $\re s>\frac12$ we have
\begin{align*}
\av
T^+F_s(z)&\=y^s\,\sum_{n=0}^\infty\biggl(\frac1{((x+n)^2+y^2)^s}\,-\,
  \int_{x+n}^{x+n+1}\frac{dt}{(t^2+y^2)^s}\biggr)
  \\
  &\qquad\qquad
  \,+\,y^s\,\int_x^0\frac{dt}{(t^2+y^2)^s}\,+\,c(s)\,y^{1-s} \,.
\end{align*}
where
$ c(s) =\int_0^\infty(t^2+1)^{-s}dt=\sqrt\pi\,\Gf(s-\frac12)/2\,\Gf(s)$.
The sum on the right converges for $\re s>0$ and the formula remains
true in this domain
(for $s\ne\frac12$) by uniqueness of meromorphic continuation. We then
have the uniform estimate
\begin{align*} \av T^+F_s(z)\,-\,c(s)\,y^{1-s} \;&\ll
y^s\sum_{n\in\ZZ}\frac{|x+n|\,+\,1}{((x+n)^2+y^2)^{s+1}}\;+\;y^s\int_x^0
y^{-2s}\,dt
 \\ & \= \oh\bigl(y^{-s}\bigr)\,+\,\oh\bigl(xy^{-s}\bigr)\,.
\end{align*}
This completes the proof for $\av T^+$. The estimate for $\av T^-$ is
exactly similar.
\end{proof}



\skipbetweenchapters
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%   Chapter 2   cocompact groups                          %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\sect{Maass forms and analytic cohomology: cocompact
groups}\label{ch-II}


In this chapter, we define a map from the space $\E_s^\Gm$ of
$\Gm$-invariant $\ld_s$-eigen\-func\-tions of the Laplace operator to
the cohomology group $H^1(\Gm;\V\om s)$. In
Section~\ref{sect-Maf-acoh} this is carried out for any discrete
$\Gm\subset \PSL_2(\RR)$. If $\Gm$ has elements of infinite order,
this map is injective. In Section~\ref{sect-co-Maf-cc} we prove that
the map is a bijection if $\Gm$ is cocompact, thus proving part of
Theorem~\ref{thmA}. As mentioned in the introduction, it is known
that $\E_s^\Gm $ and $ H^1(\Gm;\V\om s)$ are isomorphic for
cocompact~$\Gm$. Here we construct a map from cohomology to Maass
forms explicitly. This also forms a preparation for
Chapter~\ref{ch-IV}, where groups with cusps are considered.

The constructions in Section~\ref{sect-co-Maf-cc} require a
description of the cohomology using a complex based on the geometry
of the action of $\Gm$ on~$\H$. We discuss this description in
Section~\ref{sect-coh-cc}.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{From Maass forms to analytic
cohomology}\label{sect-Maf-acoh}
This section starts with a review of the standard definitions of group
cohomology. In \S\ref{ssect-Maf-coh} we construct a map
$\E^\Gm_s \lra H^1(\Gm;\V\om s)$ for any discrete
$\Gm \subset G=\PSL_2(\RR)$.


\subsubsection{Group cohomology}\label{ssect-grcoh} See, e.g.,
\cite{Brn}, Chap.~I and Chap.~III, \S1, for a general reference.



\sssect{Resolutions} For the moment let $\Gm$ denote an arbitrary
group. We recall that the homology and \indlbl{cg}{(co)homology
group}cohomology groups of $\Gm$ with coefficients in a
(right) $\QQ[\Gm]$-module $V$ are defined with help of a projective
resolution
\[ \cdots \stackrel{\p_3} \longrightarrow F_2 \stackrel
{\p_2}\longrightarrow F_1 \stackrel{\p_1}\longrightarrow F_0
\stackrel{\e}\longrightarrow \QQ\longrightarrow0\]
of the trivial $\QQ[\Gm]$-module $\QQ$ as the
(co)homology of the induced complexes
\begin{align*}
&\cdots \stackrel{ \p_2}\lra F_1 \otimes_{\QQ[\Gm]} V \stackrel {\p_1}
\lra F_0\otimes_{\QQ[\Gm]} V \lra 0\,,\\
&0\lra \Hom_{\QQ[\Gm]}(F_0,V) \stackrel {d^0}\lra
\Hom_{\QQ[\Gm]}(F_1,V)
\stackrel{d^1}\lra \cdots\,,
\end{align*}
namely\indind{$H_i(\Gm;V)$; $H^i(\Gm;V)$}{}\indlbl{dic}{$d^i$}
\be\label{chd} H_i(\Gm;V) = \ker(\partial_i)
/ \img
(\partial_{i+1})\,,\qquad
H^i(\Gm;V) = \ker(d^i)/\img(d^{i-1})\,.\ee
% De normalisatie $d^i c (x) =  (-1)^{i+1} \allowbreak c(\p_{i+1}x)$.
% wel gebruikt, maar hier niet expliciet vermeld.
We work with coefficients in $\QQ$ because this gives us more freedom
in the construction of projective resolutions
 (c.f.~\S\ref{ssect-projres}). These cohomology groups do not depend
 on the choice of the projective resolution. If $\Fpt$ and
$\tilde \Fpt$ are two projective resolutions of the trivial
$\QQ[\Gm]$-module~$\QQ$, the identity map $\QQ \lra\QQ$ can always be
lifted to an augmentation preserving chain map between the
resolutions. This lift is unique up to homotopy, and induces
isomorphisms of the homology and cohomology groups determined by the
resolutions. The construction of such a chain map
$\Fpt\lra \tilde \Fpt$ may depend on many choices, so it may take
work to describe the corresponding isomorphism of the
(co)homology groups explicitly. For this reason it is important for
explicit cohomological constructions to choose a specific resolution
with good properties.

\sssect{Standard resolution} The
\indlbl{sm}{standard model of group cohomology}standard model of group
cohomology is obtained from the \emph{standard resolution}
$\Fpt^\gcoh$, where \indlbl{Fgc}{standard resolution;
$F_i^\gcoh$}$F_i^\gcoh $ is the free $\QQ[\Gm]$-module
$ \QQ[ \Gm^{i+1}]$. The
\indlbl{pibd}{boundary map, $\p_i$}boundary maps $\partial_i$, 
the
\indlbl{augm}{augmentation; $\e$}augmentation $\e$, and the
$\Gm$-action are induced by
\be\label{st-res}\begin{aligned}
\p_i(\gm_0,\cdots,\gm_i) &\= \sum_{j=0}^i
(-1)^j
(\gm_0,\cdots,\widehat{\gm_j},\cdots,\gm_i)\,,\\
\e(\Gm_0)&\=1\,,\qquad
(\gm_0,\cdots,\gm_i)\,|\,\gm \=
(\gm_0\gm,\cdots,\gm_i\gm)
\,.
\end{aligned}\ee

In this model, an
\indlbl{cch}{cochain}$i$-cochain is represented by a $\Gm$-equivariant
map $c:\Gm^{i+1}\lra V$, which is then extended by linearity to
$c: F^\gcoh_i \lra V$. The equivariance implies that $c$ is
completely determined by its restriction $\ps$ to
$\Gm^i\times\{1\}\subset \Gm^{i+1}$, and one often uses this
inhomogeneous version. (The last variable is then omitted from the
notation and the definition of the coboundary map is modified in the
obvious way.)

Each $F^\gcoh_i$ is a free $\QQ[\Gm]$-module. In dimension $i>0$ the
rank is infinite if $|\Gm|=\infty$. In dimension~$i=0$, the cocycles
satisfy $c(\gm) = c(1)v\,|\,\gm$, and hence are determined by
$c(1)\in V^\Gm$, and since there are no cochains in dimension $-1$ we
have
\be H^0(\Gm;V) \= V^\Gm \= \left\{v\in V\;:\; v\,|\,\gm=v\text{ for
all }\gm\in \Gm \right\}\,.\ee
For homology we find that $H_0(\Gm;V) = V_\Gm$, where
\indlbl{coinv}{coinvariants; $V_\Gm$}$V_\Gm $ is the submodule of
\emph{coinvariants}
$ V / \langle v\,|\,(1-\gm)\;:\; v\in V,\, \gm\in \Gm\rangle$.

In dimension $i=1$ the standard model gives
\indlbl{hc}{homogeneous cocycle}homogeneous cocycles
$(\gm_0,\gm_1)\mapsto c(\gm_0,\gm_1)\in V$ satisfying for all
$\gm_j\in\Gm$:
\[ c(\gm_0,\gm_1)\,|\,\gm_2\=c(\gm_0\gm_2,\gm_1\gm_2)\quad\text{ and
}\quad c(\gm_0,\gm_1)+c(\gm_1,\gm_2) = c(\gm_0,\gm_2)\,.\]
Such a $1$-cocycle is a
\indlbl{cb}{coboundary}coboundary if
$c(\gm_0,\gm_1) = f(\gm_0)-f(\gm_1)$ for some $f: \Gm\lra V$
satisfying $f(\gm_1\gm_2)=f(\gm_1)\,|\,\gm_2$. Going over to
\indlbl{ic}{inhomogeneous cocycle}inhomogeneous cocycles
$\gm\mapsto \ps_\gm=c(\gm,1)$, we get the following well-known
description of the first cohomology group:\indind{ $Z^i(\Gm;V)$;
$B^i(\Gm;V)$}{}
\be\label{H1def}
\begin{aligned}
H^1( \Gm;V) &\= Z^1( \Gm;V) / B^1( \Gm;V)\,,\\
Z^1( \Gm;V)&\= \left\{ \ps:\Gm\lra V\;:\;
\ps_{\gm\dt}=\ps_\gm\,|\,\dt+\ps_\dt\text{ for all }\gm,\dt\in \Gm
\right\}\,,\\
 B^1( \Gm;V) &\= \left\{ \gm \mapsto v\,|\,(\gm-1)\;:\; v\in
 V\right\}\,.
 \end{aligned}
\ee

\subsubsection{From invariant eigenfunctions to
cohomology}\label{ssect-Maf-coh}
Let $\Gm\subset G$ be an arbitrary subgroup, provided with the
discrete topology. We now shall explicitly define a linear map
$\bold r:\E_s^\Gm\lra H^1(\Gm;\V\om s)$, and show that it is
injective if $\Gm$ has elements of infinite order.

\sssect{Definition of analytic cocycles associated to invariant
eigenfunctions}Let $u\in \E^\Gm_s$. We associate to it three
inhomogeneous cocycles $(r_\gm) \in Z^1(\Gm;\V\om s)$,
$(p_\gm) \in Z^1(\Gm;\E^\om_s)$ and $(q_\gm) \in Z^1(\Gm;\W\om s)$
which correspond to one another under the isomorphisms
$\V\om s \cong \E^\om_s \cong \W\om s$ given by the Poisson
transformation and the transverse Poisson transformation. These
cocycles are obtained by integrating the $\Gm$-invariant closed
$1$-forms
\[ [u,R(\z;\dv)^s]\,,\qquad [u,p_s(\dv,z)]\,,\qquad
[u,q_s(\dv,z)]\,.\]
The $\Gm$-invariance follows from \eqref{[]invar}, \eqref{Requiv} and
\eqref{pq-Ginv}. We choose a
\indlbl{bp}{base point}base point $z_0\in \H$, and integrate over a
path from $\gm^{-1}z_0$ to $z_0$:\indind{$(r_\gm)$; $(p_\gm)$;
$(q_\gm)$}{}
\beqabclbl{pqrdef}
\begin{align}
\label{rdef}
r_\gm(\z) &\= \int_{\gm^{-1}z_0}^{z_0} [u,R(\z;\dv)^s]\,,
\displaybreak[0]\\
\label{pdef}
p_\gm(z) &\= \int_{\gm^{-1}z_0}^{z_0} [u,p_s(\dv,z)]\,,
\displaybreak[0]\\
\label{qdef}
q_\gm(z) &\= \int_{\gm^{-1}z_0}^{z_0} [u,q_s(\dv,z)]\,.
\end{align}
\eeqabc
So $r_\gm\in \V\om s$, $p_\gm\in \E_s$. We identify $q_\gm$ with the
element of $\W\om s$ represented by it. Changing the choice of the
base point changes the cocycles by a coboundary. Thus, we find
respectively \indlbl{pqr}{$\bold p$; $\bold q$;
$\bold r$}$\bold r u \in H^1(\Gm;\V\om s)$,
$\bold p u \in H^1(\Gm;\E^\om_s)$ and
$\bold q u \in H^1(\Gm;\W\om s)$, which depend linearly on~$u$.
Relations \eqref{Poi-R} and \eqref{ex1c} imply that
\be\label{pqr-rels} \bold p \= \poiss s \bold r\,,\qquad \bold q =
\bb(s) \tpoiss s \bold r\,, \ee
with $\bb(s)$ as in~\eqref{btd}.

The Poisson kernel $R$ in \eqref{rdef} is considered to be general.
Strictly speaking, \eqref{rdef} defines the cocycle
$\gm\mapsto r_\gm$ in the line model of~$\V\om s$. In the projective
model, work with $R^\prj$, and in the circle model with $R^\ci$ and a
base point in~$\disk$. See~\eqref{R-othermodels}.

\sssect{Symmetry $s\leftrightarrow 1-s$} Since
$\E_s^\om=\E_{1-s}^\om$, we can carry out the construction with $s$
replaced by $1-s$. Denote the corresponding cocycles by $\hat r_\gm$,
$\hat p_\gm$ and $\hat q_\gm$. For $s\neq \frac12$ we have
$\hat r_\gm = I_s r_\gm$, $\hat p_\gm=p_\gm$, by \eqref{Poi-R}
and~\eqref{IR0}, and $p_\gm = \frac{\tan\pi
s}\pi\bigl( q_\gm -\nobreak \tilde q_\gm\bigr)$, by \eqref{ex1c}
and~\eqref{splitting}.
\be
\begin{aligned}& \xymatrix{
& H^1\left( \Gm;\V\om{1-s}\right)\ar[r]^{\raisebox{1ex}{$\scriptstyle 
\bb(1-s)\,\tpoiss {1-s}$}} \ar[ld]_{\poiss {1-s}}
& H^1\left(\Gm;\W\om{1-s}\right)
\\ H^1\left( \Gm;\E^\om_s\right)
& \E^\Gm_s\ar[u]_{\hat{\bold r}}\ar[ru]^{\hat{\bold q}}\ar[l]_{\bold
p} \ar[d]^{\bold r}\ar[rd]^{\bold q}
\\
& H^1\left(\Gm;\V\om s\right)
\ar[r]^{\raisebox{.3ex}[1ex][0ex]{$\scriptstyle \bb(s)\,\tpoiss s$}}
\ar[lu]_{\poiss s} \ar@/_6mm/[uu]_{I_s}
& H^1\left( \Gm;\W\om s\right) }
\end{aligned} \ee

\sssect{Formulation with hyperfunctions} Another point of view uses
the isomorphism $\E_s^\Gm \cong (\V{-\om}s)^\Gm$ induced by the
Poisson transformation. The short exact sequence \eqref{hdefG}
induces a long exact sequence of cohomology groups, which gives a
connecting homomorphism
$(\V{-\om}s)^\Gm = H^0(\Gm;\V{-\om}s) \lra H^1(\Gm;\V\om s)$. To
describe this map explicitly, we choose for a given $\al \in
(\V{-\om}s)^\Gm$ a representative $g\in \HHH_s$. This gives the
inhomogeneous cocycle $\gm\mapsto \ps_\gm = g\,|\,(\gm-\nobreak1)$
with values in $\V\om s$.

For a given $u\in \E^\Gm_s$ let $\al \in (\V{-\om}s)^\Gm$ be chosen
such that $\poiss s\al=u$. Theorem~\ref{thmPoiss-inv} gives an
explicit choice
(depending on $u$ and a base point $z_0\in \uhp$) for a
representative~$g$ of~$\al$, in the projective model of the principal
series. Using the same base point in \eqref{rdef}, we find for
$\z\in \uhp$ near $\proj\RR$ and $\gm\in\Gm$:
\begin{align*}
\ps^\prj_\gm (\z) &\= u(z_0) \, R^\prj(\dv;z_0)^s|^\prj_{2s}\,\gm
\,(\z)
+ \int_{z_0}^{\gm \z}
\bigl[u(z'),R^\prj(\dv;z')^s|^\prj_{2s}\,\gm\,(\z)
\bigr]_{z'}
\\
&\qquad\hbox{}
- u(z_0) \, R^\prj(\z;z_0)^s
- \int_{z_0}^\z \bigl[u,R^\prj(\z;\dv)^s\bigr]
\displaybreak[0]
\\
&\= u(z_0) \, R^\prj(\dv;z_0)^s|^\prj_{2s}\,(\gm-1)\, (\z)
+ \int_{\gm^{-1}z_0}^\z \bigl[u, R^\prj(\z;\dv)^s\bigr] -
\int_{z_0}^\z\bigl[u,R^\prj(\z;\dv)^s\bigr]
\displaybreak[0]\\
&\= u(z_0) \, R^\prj(\dv;z_0)^s|^\prj_{2s}\,(\gm-1)\, (\z)
+r^\prj_\gm(\z)\,,
\end{align*}
where we have used \eqref{[]invar}, \eqref{Requiv} and the
$\Gm$-invariance of~$u$. For $\z$ in the lower half-plane, near
$\proj\RR$, we also use \eqref{[]def} and \eqref{{}alt} to find the
following equality.
\begin{align*}
\ps^\prj_\gm(\z)&\= \int_{\overline{\gm\z}}^{z_0} \bigl[
R^\prj(\dv;z')^s|^\prj_{2s}\,\gm\,(\z),u(z')\bigr]
- \int_{\bar \z}^{z_0} \bigl[R^\prj(\z;\dv)^s,u\bigr]
\displaybreak[0]\\
&\= \biggl( \int_{\bar \z}^{\gm^{-1}z_0} - \int_{\bar \z}^{z_0}
\biggr)\, \biggl(
-\bigl[u,R^\prj(\z;\dv)^s\bigr]+ d\bigl(uR^\prj(\z;\dv)^s\bigr)
\biggr)
\displaybreak[0]\\
&\= r^\prj_\gm(\z) + u(\gm^{-1}z_0) \, R^\prj(\z;\gm^{-1}z_0)^s -
u(z_0)
\, R^\prj(\z; z_0)^s
\displaybreak[0]\\
&\= r^\prj_\gm(\z) + u(z_0) \;
R^\prj(\dv;z_0)^s|^\prj_{2s}\,(\gm-1)\,(\z)\,,
\end{align*}
which is the same expression as we obtained for $\z\in \uhp$. By
holomorphic continuation this description also holds for
$\z\in \proj\RR$. Thus, the cocycle $\gm\mapsto \ps_\gm$ represents
the same cohomology class $\bold r u$ as $\gm\mapsto r_\gm$.\medskip

\begin{prop}\label{prop-rinj}If the discrete subgroup $\Gm\subset G$
is infinite, then $\bold r$, $\bold p$ and $\bold q$ are injective.
\end{prop}
\begin{proof}
For the injectivity it suffices to consider only $\bold r$, since
$\bold p$ and $\bold q$ are isomorphic transforms of~$\bold r$. The
formulation with hyperfunctions shows that $\bold r$ corresponds to
the connecting homomorphism $\dt$ in the part
\[ \lra \HHH_s^\Gm \lra (\V{-\om}s)^\Gm \stackrel\dt\lra H^1(\Gm;\V\om
s)
\lra\]
of the long exact sequence associated to \eqref{hdefG}. Hence it
suffices to show that $\HHH_s^\Gm=\{0\}$.

We use the circle model. Let $g=g^\ci \in \HHH_s^\Gm\sm \{0\}$.
Interchanging if necessary the roles of the interior and exterior of
$\uc$, we can assume that $g$ is holomorphic and non-zero on the
annulus $R=\{c\leq |w|<1\}$ for some $c <1$. Then the $1$-form
$\om = d\log g = \frac{g'(w)}{g(w)}\, dw$ is meromorphic on $R$, with
integral residues. Since $\disk\sm R$ is compact and $\Gm$ is
infinite and discrete, we can choose
$\gm = \matc A B {\bar B} {\bar A} \in \Gm$ such that
$R\cup\gm R =\disk$. The $\Gm$-invariance implies
$g(\gm w)=g(w)  \bigl( (\bar A+\nobreak
\bar B w)(A+\nobreak B w^{-1})\bigr)^s$, and hence
\[ \om \circ \gm - \om \= s\, \Bigl( \frac 1{w+\bar A/\bar B}\,-\,
\frac 1w\,+\,\frac1{w+B/A} \Bigr)\]
near $\p \disk$. It follows that $\om$ has a meromorphic continuation
to all of $ \disk$ and has integral residues everywhere except at
$w=0$, where its residue equals $s$.
(Note that the point $-B/A=\gm^{-1}0$ lies in $R$.)
This is a contradiction since $\frac 1{2\pi i} \int_{|w|=c}\om =
\frac1{2\pi i}\int_{|w|=c} d\log g\,\in\,\nobreak \ZZ$ and
$s\not\in\ZZ$.
\end{proof}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Cohomology for cocompact groups}\label{sect-coh-cc}


The description of group cohomology in \S\ref{ssect-grcoh} with the
standard resolution $\Fpt^\gcoh$ does not use the fact that $\Gm$
acts on the hyperbolic plane. We will mention in
\S\ref{ssect-projres} and \S\ref{ssect-cc-ex} several resolutions of
geometrical nature, and describe group cohomology with these
resolutions in~\S\ref{ssect-grcoh-g}. There we also formulate the
linear maps $\bold r$, $\bold p$ and $\bold q$ in terms of these
geometrical resolutions.



\subsubsection{Projective resolutions of geometric
nature}\label{ssect-projres}


Group cohomology can be computed based on a free action of the group
on a contractible set. Here we consider
\indlbl{cc}{cocompact}cocompact discrete
\indlbl{gam}{$\Gm$}$\Gm\subset G$, i.e., discrete subgroups for which
the quotient $\Gm\backslash\H$ is compact. The space $\H$ is
contractible. However the action of $\Gm$ on $\H$ is not free if
$\Gm$ has elliptic elements, which have finite order. We circumvent
this problem by working only with $\Gm$-modules that are vector
spaces over~$\QQ$.

First we discuss a resolution that is similar to $\Fpt^\gcoh$ in
\S\ref{ssect-grcoh}, but rather large. More practical are smaller
resolutions, based on a $\Gm$-tesselation of~$\H$.


\sssect{Chain complex on $\H$} The action of $\Gm$ on the contractible
space $\H$ is taken into account in the complex $\Fpt^\hcoh$ defined
by \indlbl{fhyp}{chain complex on $\H$; $\Fpt^\hcoh$; $\p_i$;
$\e$}$F^\hcoh_i = \QQ[\H^{i+1}]$, with boundary maps $\p_i$,
augmentation $\e$, and group action determined by
\be\label{chH}
\begin{aligned}
\p_i (P_0,\cdots,P_i) &\= \sum_{j=0}^i
(-1)^j(P_0,\cdots,\hat
P_j,\cdots,P_i)\,,\\
\e (P_0)&\=1\,,\\
(P_0,\cdots,P_i)\,|\,\gm&\=
(\gm^{-1}P_0,\cdots,\gm^{-1}P_i)\,.
\end{aligned}\ee
In low dimensions, we can think of the generators of $\QQ[\H^{i+1}]$
as geometric objects: $(P)$ corresponds to the point~$P\in\H$, and
$(P,Q)$ corresponds to the geodesic segment oriented from $P$ to $Q$
(degenerate if $P=Q$). The generator $(P,Q,R)$ corresponds to a
(possibly degenerate) triangle with a numbering of it vertices.


The $\QQ[\Gm]$-modules $\QQ[\H^{i+1}]$ need not be free if $\Gm$ has
elliptic elements, which fix points in~$\H$. To see that
$\QQ[\H^{i+1}]$ is a projective $\QQ[\Gm]$-module, we have to show
that there is a $\QQ[\Gm]$-linear lift $s: \QQ [\H^{i+1}] \lra B$ for
each given $t: \QQ[\H^{i+1}] \lra C$ in each exact sequence of
$\QQ[\Gm]$-modules:
\be \raisebox{.8cm}{\xymatrix{ 0\ar[r]&A\ar[r]& B\ar[r]& C\ar[r]& 0\\
&&\QQ[\H^{i+1}]\ar[u]^s\ar[ru]_t }}\ee
This is done by taking lifts $b_x\in B$ of $t(x)\in C$ for a set of
$x\in \H^{i+1}$ generating the $\QQ[\Gm]$-module $\QQ[\H^{i+1}]$, and
determining $s$ by
$s(x)=\frac1{|\Gm_x|} \sum_{\gm\in\Gm_x} b_x \,|\, \gm$, where
$\Gm_x$ is the stabilizer of $x$ in~$\Gm$. Then $s$ can be extended
$\QQ[\Gm]$-linearly. See \cite{Brn}, Chap.~I, \S8, for a further
discussion of projective modules.

There are augmentation preserving chain maps
$\Fpt^\gcoh \lra \Fpt^\hcoh$ and $\Fpt^\hcoh \lra \Fpt^\gcoh$ that
induce isomorphisms in the cohomology groups. The latter requires
uncountably many choices to be made. The former can be obtained with
only one choice: Take a base point $P_0\in \H$. This leads to the
explicit chain map induced by $(\gm_0,\cdots,\gm_i) \mapsto
(\gm_0^{-1}P_0,\cdots,\gm_i^{-1}P_0)$.

This description of (co)homology, with the chain complex on~$\H$ as
the projective resolution, can be used to describe group cohomology
for any discrete subgroup $\Gm\subset G$ for which the isotropy
groups of elements of $\H$ are finite.
\smallskip

The symmetric group $\mathfrak S_{i+1}$ acts on $F^\hcoh_i$ by
$\pi (P_0,\ldots,P_i) = (P_{\pi 0},\ldots, P_{\pi i})$. Let
\indlbl{Fhyp-}{$\Fpt^{\hcoh-}$}$F^{\hcoh -}_i$ be the subspace on
which $\mathfrak S_{i+1}$ acts by the sign character. These spaces
form a subcomplex $\Fpt^{\hcoh-}\subset \Fpt^\hcoh$, which is also a
projective resolution of~$\QQ$. A chain map
$ \Fpt^\hcoh\lra \Fpt^{\hcoh-}$ is the \indind{antisymmetrization;
$A_\pnt$}{}\emph{antisymmetrization} $A_\pnt$ determined by
\be\label{antisym} A_i: (P_0,\ldots,P_i)
\mapsto \frac1{(i+1)!}\sum_{\pi \in \mathfrak S_{i+1}} \sign(\pi) \,
(P_{\pi 0},\ldots, P_{\pi i})\,. \ee
This variant $\Fpt^{\hcoh-}$ (in which, for example, $(P,Q,R)$ now
corresponds to a triangle which is still oriented, but no longer has
a numbering of its vertices) is often more convenient than the
resolution $\Fpt^\hcoh$ itself. (One could avoid introducing
denominators by defining $\Fpt^{\hcoh-}$ as a quotient complex rather
than a subcomplex of $\Fpt^{\hcoh}$, but in any case they do not
disturb us since we work over~$\QQ$.)


\sssect{Resolutions based on a tesselation}The models that we like
best are geometrical, and finite in two ways: Each $F_i$ is finitely
generated as a $\QQ[\Gm]$-module, and $F_i$ vanishes for $i>2$. Here
we use that $\Gm \subset G$ is discrete and cocompact.


By a
\indlbl{te}{tesselation}\emph{tesselation} we mean a locally finite
$\Gm$-invariant covering $\tess$ of $\H$ by compact polygons with
geodesic boundary segments. The polygons overlap at most in their
boundaries. Such a covering gives rise to the set
\indlbl{pol}{polygon}\indlbl{Xj}{$X_j$, $X_j^\tess$}$X_2=X_2^\tess$ of
polygons of~$\tess$
(with the orientation inherited from that of~$\H$), the set
$X_1=X_1^\tess$ of oriented \indlbl{edg}{edge}edges of~$\tess$
(with each element of $X_1$ arising as a boundary component of two
neighboring elements of~$X_2$), and the set $X_0=X_0^\tess$ of
\indlbl{vert}{vertex}vertices. For vertices $P$ we have $\e (P)=1$.
Each $e\in X_1$ is the oriented edge
\indlbl{ePQ}{$e_{P,Q}=e(P,Q)$}$e_{P,Q} $ (or $e(P,Q)$ when we want to
avoid subscripts)
joining some vertex $P$ of~$\tess$ to a neighboring vertex~$Q$. Thus
$e_{Q,P}=-e$ in $\QQ[X_1]$, so that we have chosen only one of the
two possible orientations of the edge in defining $X_1$. We then
define $\p_1:\QQ[X_1]\lra \QQ[X_0]$ by $\p_1 e =
(Q)-(P)$. A polygon $V\in X_2$ has vertices $P_1, \cdots,
P_l$, ordered corresponding to the orientation of $\H$. The boundary
is $\p_2 V = e_{P_1,P_2}+e_{P_2,P_3}+ \cdots+
e_{P_{l-1},P_l}+e_{P_l,P_1}$. The $\Gm$-action is induced by
$P\mapsto \gm^{-1}P$ in~$\H$. In this way, we have for each
tesselation $\tess$ a resolution
\indlbl{F-Tcal}{$\Fpt^\tess$}$\Fpt^\tess$:
\be 0 \longrightarrow \QQ[X_2] \stackrel{\p_2}\longrightarrow \QQ[X_1]
\stackrel{\p_1}\longrightarrow \QQ[X_0] \stackrel{\e}\longrightarrow
\QQ\longrightarrow 0\,\ee
of the trivial $\QQ[\Gm]$-module $\QQ$. So $F^\tess_i= \QQ[X_i]$ for
$i=0$, $1$, $2$, and $F^\tess_i=\{0\}$ otherwise. As above, we can
check that the $\QQ[X_i]$ are projective.

If $f=\sum_l \al_l x_l \in F^\tess_i$ with $x_l\subset B$ for all $l$
for some subset $B\subset\H$, then we say that $f$ is
\emph{supported} in $B$. The
\indlbl{su}{support of chain; $\supp$}\emph{support} $\supp f$ of~$f$
is the intersection of all such~$B$. It is a compact subset of~$\H$,
or empty if $f=0$.

It is clear that $\Fpt^\tess$ is a complex. We check the exactness. We
pick a base point $\tilde P \in X_0$. Suppose that
$f=\sum_{P\in X_0} a_P  \, (P)$ is in the kernel of~$\e$. If
$f\neq0$, take $P\in \supp f$ with maximal distance to~$\tilde P$,
where the distance is computed along edges in~$X_1$. If there is a
neighbor $Q$ of $P$ with smaller distance to~$\tilde P$ subtract
$\p_1 \bigl(a_P \,  e_{Q,P}\bigr)$ from $f$. This removes $P$ from
the support of~$f$. Otherwise, choose an ordering of the points with
maximal distance to~$\tilde P$, and remove them successively. (Each
of them has a neighbor that has at most the same distance
from~$\tilde P$.)
This process stops when $f=0$, or $\supp f \subset \{\tilde P\}$, and
then $\e f=0$ also implies $f=0$.

We call a
\indlbl{1ch}{$1$-chain}\emph{$1$-chain} an element
$C= \sum_{e\in X_1} a_e  \, e \in \ZZ[X_1]$; so all $a_e\in \ZZ$. We
may view $C$ as a
(possibly non-connected) path in $\H$ along edges of~$\tess$, with
integral multiplicities. If $\p_1 C=0$, we call it a
\indlbl{1cy}{$1$-cycle}\emph{$1$-cycle}. In this case, the number of
times that a point $P\in X_0^\tess$ occurs as the terminal point of
an edge in~$C$
(counted with multiplicities) is equal to the number of times that it
occurs as the initial point of an edge in~$C$. Thus, a $1$-cycle
corresponds to a combination of closed paths along edges
in~$X_1^\tess$. To each $z\in \H$ not on the edges in~$C$ is
associated a \indlbl{wn}{winding number; $m_C$}\emph{winding number}
$m_C(z)\in \ZZ$ of $C$ around~$z$. This function $m_C$ has bounded
support, and is constant on the interior of the polygons in
$X_2^\tess$. The $2$-chain
$D= \sum_{V\in X_1} m_C(V) \, V\in \ZZ[X_2^\tess]$ satisfies
$\p_2 D=C$. In particular, the kernel of
$\p_1: \ZZ[X_1] \lra \ZZ[X_0]$ is equal to $\p_2 \ZZ[X_2]$. Since
$\ZZ[X_i]$ has no torsion, tensoring with $\QQ$ gives the exactness
of $\Fpt^\tess$ at dimension~$1$.

If $f=\sum_{V\in X_2} a_V\, V\in \QQ[X_2]$ is non-zero, consider an
edge $e$ in the boundary of $\supp f$. So $e$ is a boundary segment
of only one $V\in X_2$, and $a_V\neq 0$. In $\p_2 f$, this edge
occurs with coefficient $ a_V$ or $-a_V$. This shows that $\p_2$ is
injective.\medskip


If the tesselation $\Scal$ is a
\indlbl{ref}{refinement}refinement of the tesselation $\tess$, there
is an augmentation preserving chain map
$\Fpt^{\tess} \lra \Fpt^{\Scal}$, where each $x\in X_i^\tess$ is
mapped to the sum of the $y\in X_i^\Scal$ into which it is
subdivided. This induces an isomorphism in homology and cohomology.
Since any two tesselations have a common refinement, this permits an
explicit identification of the
(co)homology groups constructed using the resolutions coming from
distinct tesselations.

A
\indlbl{tr}{triangulation}\emph{triangulation} is a tesselation for
which all polygons $V\in X_2$ are triangles. Any tesselation can be
refined to a triangulation. By
\indlbl{tria}{$\Dt(\dv,\dv,\dv)$}$\Dt(P,Q,R)$ we denote the triangle
with vertices $P$, $Q$ and $R$ ordered by the positive orientation
of~$\H$.

Let $\tess$ be a $\Gm$-invariant triangulation. An augmentation
preserving chain map $a_\pnt: \Fpt^\tess \lra \Fpt^{\hcoh-}$ can be
defined by\indind{$a_\pnt$}{}
\be\label{a-def}
\begin{aligned}
a_0(P) &\= (P)&\quad &(P\in X_0)\,,\\
a_1 e_{P,Q} &\= \frac12(P,Q)-\frac12(Q,P)&\quad&(e_{P,Q}\in X_1)\,,\\
a_2 \Dt(P,Q,R) &\= \frac 16 \sum_{\pi\in \mathfrak S_3} \sign(\pi)
 \, (\pi P,\pi Q,\pi R)&\quad&\Dt(P,Q,R)\in X_2)\,.
\end{aligned}
\ee
Since every tesselation can be refined to a triangulation, this gives
explicit chain maps between any $\Fpt^\tess$ and $\Fpt^\hcoh$.

Next we discuss an augmentation preserving chain map
$f_\pnt: \Fpt^\gcoh \lra \Fpt^\tess$ for any tesselation~$\tess$.
Choose a base point $P_0\in X_0^\tess$, and define
$F^\gcoh_0\lra F^\tess_0$ by $f_0(\gm) = \gm^{-1} P_0$. In
dimension~$1$, choose for each $\gm\in\Gm$ a path $p_\gm\in\ZZ[X_1]$
from $P_0$ to $\gm^{-1}P_0$ along edges in $X_1^\tess$, and extend
the definition $f_1(1,\gm) = p_\gm$ to $F_1^\gcoh$ in a
$\QQ[\Gm]$-linear way. For $(1,\gm,\dt) \in F^\gcoh_2$, the sum
$C= f_1(1,\gm)+f_1(\gm,\dt)+f_1(\dt,1)$ is a $1$-cycle in
$\ZZ[X_1^\tess]$. Take $D\in \ZZ[X_2^\tess]$ such that $\p_2 D = C$.
The map $f_2$ is determined by $f_2(\e,\gm\e,\dt\e) = \e^{-1} D$. For
$i>2$, put $f_i=0$.\medskip

The resolutions coming from tesselations and the chain maps between
them work with coefficients in~$\ZZ$ instead of~$\QQ$. We need the
order of elliptic elements as denominators for the projectivity, and
we have used the denominators $2$ and $3$ in the construction of the
chain map $a_\pnt:\Fpt^\tess \lra \Fpt^\hcoh$.



\subsubsection{Choices of tesselations}\label{ssect-cc-ex}We shall use
four special types of tesselations in particular, leading to four
models of cohomology.
\begin{description}
\item[Tesselations of type \Fcoh]\indlbl{typeF}{type \Fcoh} Let
\indlbl{fd}{fundamental domain; $\fd$}$\fd $ be a connected closed
\emph{fundamental domain} of $\Gm\backslash \H$ with finitely many
geodesic sides. Use the tesselation $\tess$ with the $\Gm$-translates
$\gm\fd$ as the set of polygons
$ X_2 =X_2^\tess=\{ \gm \fd\;:\; \gm\in\Gm \}$. If an elliptic fixed
point occurs at the center of an edge $e$ in $\p\fd$, we add this
point to $X_0$, and divide the edge~$e$ into two edges.

In the resulting resolution, $\QQ[X_2]$ is a free $\QQ[\Gm]$-module of
rank one, with basis $(\fd)$. The fundamental domain has an even
number $2n$ of of edges in its boundary. There exists a set
\indlbl{Ee}{$E$}$E$ of $n$ of these edges and a set of generators
$\left\{ \gm_e\;:\; e\in E\right\}$ of $\Gm$ such that
$\p_2\fd = \sum_{e\in E} \bigl(e-\gm_e^{-1}e\bigr)$. The module
$\QQ[X_1]$ is a free $\QQ[\Gm]$-module with the $e\in E$ as a basis.
If $\Gm$ has elliptic elements, then $\QQ[X_0]$ is not free.

\item[Tesselations of type \Dcoh]\indlbl{typeD}{type \Dcoh}Fix a point
$P_0\in \H$ that is not an elliptic fixed point of~$\Gm$. Form the
\indlbl{Di}{Dirichlet fundamental domain}\emph{Dirichlet fundamental
domain} $\fd$ consisting of all points $P\in \H$ for which
$\dist(P,P_0) \leq \dist(\gm P,P_0)$ for all $\gm\in \Gm$. This
fundamental domain $\fd$ shares sides with finitely many translates
$\al ^{-1}\fd$, $\al\in \Gm$. These $\al$'s form a finite set
\indlbl{A}{$A$}$A=A^{-1}$ generating~$\Gm$. We take $X_0=\Gm \, P_0$,
and $X_1 = \bigl\{ \gm^{-1} e_{P_0,\al^{-1} P_0} \;:\; 
\gm\in \nobreak \Gm;\, \al\in \nobreak A\bigr\}$. The elements of
$X_1$ divide $\H$ into polygons indexed by the $\Gm$-orbits of the
vertices of $\fd$. This tesselation is dual to the tesselation of
type \Fcoh\ for the same Dirichlet fundamental domain. $\QQ[X_0]$ is
free on the generator $(P_0)$. The fundamental domain $\fd$ is not
necessarily the union of elements of~$X_2$.

\item[Tesselations of type \Mcoh]\indlbl{typeM}{type \Mcoh}Start with
a tesselation $\tess$ of type \Fcoh\ for a Dirichlet fundamental
domain $\fd$ with a base point $P_0$ in the interior of~$\fd$. We add
to $X_0^\tess$ the $\Gm$-translates of~$P_0$, and to $X_1^\tess$ the
$\Gm$-translates of the edges from $P_0$ to the vertices of~$\fd$. We
call the resulting refinement $\Scal$ of $\tess$ a tesselation of
type
\indlbl{mcoh}{\Mcoh}\Mcoh. It is a triangulation.

\item[Tesselations of type \Mcoha]\indlbl{typeMa}{type \Mcoha}This is
a further refinement of a triangulation of type
\indlbl{mcoha}{\Mcoh}\Mcoh. We add the $\Gm$-translates of the
intersection points of the geodesic segments from $P_0$ to
$\al^{-1}P_0$, with $\al\in A$ as above, and the sides of~$\fd$. We
also add the $\Gm$-translates of the resulting edges from $P_0$ to
the new points. The resulting refinement is a triangulation.
\end{description}

\sssect{Fundamental class}Let $\tess$ be a tesselation of type \Fcoh.
Since $V \otimes_\QQ\QQ = V_\Gm$, for the trivial
$\QQ[\Gm]$-module~$\QQ$, we have
\[ H_2(\Gm;\QQ) \;\cong\; \ker\bigl(
(F^\tess_2)_\Gm \lra
(F^\tess_1)_\Gm\bigr) \=(F^\tess_2)_\Gm \;\cong\; \QQ\,,\]
with the class of $(\fd)$ in~$(F^\tess_2)_\Gm$ as its generator. This
element of $H_2(\Gm;\QQ)$ is the
\indlbl{fc}{fundamental class; $[\Gm\backslash\H]$}\emph{fundamental
class}. We denote it by $[\Gm\backslash\H]$. If the fundamental
domain $\fd$ on which $\tess$ is based is a Dirichlet fundamental
domain, we have the chain map constructed above to the resolution
$\Fpt^\Scal$ for the refinement $\Scal$ of $\tess$ of type~\Mcoh\
or~\Mcoha. This chain map induces an isomorphism in homology. The
fundamental class is also represented by
$(\fd)=\sum_{V\in X_2^\Scal,\, V \subset\fd}  (V)$ in the
descriptions of these types. In the description of type \Dcoh, the
fundamental class is not represented by an element of $\ZZ[X_2]$.

\subsubsection{Cocycles}\label{ssect-grcoh-g}
In each of the models, the group cohomology of~$\Gm$ with values in a
right $\QQ[\Gm]$-module~$V$ is obtained up to isomorphism as the
cohomology
\indlbl{fco}{cohomology group}\indlbl{fcoc}{cohomology group,
$C^i(\Fpt;V)$} of the complex
$C^\hpnt(\Fpt;V) = \Hom_{\QQ[\Gm]}(\Fpt,V)$. If $F_i=\QQ[X_i]$, this
is equal to the complex $\map(X_\pnt,V)^\Gm$, where we define
\indlbl{map}{$\map(X,V)$}$\map(X_i,V)$ as the $\QQ$-linear space of
all functions $X_i\lra V$, with the action
$f^\gm(x) = f(\gm x)\,|\,\gm$. We denote by
\indlbl{fcoz}{cohomology group, $Z^i(\Fpt;V)$}$Z^i(\Fpt;V)$ the kernel
of $d^i : C^i(\Fpt;V)\lra C^{i+1}(\Fpt;V)$, and by
\indlbl{fcob}{cohomology group, $B^i(\Fpt;V)$}$B^i(\Fpt;V)$ the image
$d^{i-1} \, C^{i-1}(\Fpt;V)$.


\sssect{Dimension $0$} In all models, it is easy to check that
$H^0(\Gm;V) = V^\Gm$.

\sssect{Dimension $1$} In the model based on a chain complex on~$\H$,
the $1$-cochains are the maps $c: \H^2\lra V$ that satisfy
\be \label{c-Gi}c(\gm^{-1}P,\gm^{-1}Q) = c(P,Q)\,|\,\gm
\qquad(\text{$\Gm$-equivariance})\,.\ee
For $c$ to be a $1$-cocycle, the additional condition
is\indind{cocycle relation}{}
\be \label{c-co}
c(P,Q)+c(Q,R)=c(P,R)
\qquad (\text{cocycle relation)}\,.\ee
This implies that $c(P,P)=0$ and that $c(Q,P)=-c(P,Q)$. The
$1$-cocycle~$c$ is a $1$-coboundary if $c(P,Q) = f(P)-f(Q)$ for some
$\Gm$-equivariant map $f:\H\lra V$.


In models built from a tesselation, the description is similar. A
$1$-cochain is determined by an equivariant map $c:X_1\lra V$. We can
define $c(p)$ for any path~$p$ along edges in~$X_1$ by linearity. If
$c$ is a cocycle, then $c(p)$ depends only on the end points of~$p$,
so we get a map $c: X_0^2\lra V$ satisfying \eqref{c-Gi}
and~\eqref{c-co}. There is always a map $f:X_0 \lra V$ such that
$c(P,Q)=f(P)-f(Q)$.
(Choose $f(P_0)\in V$ arbitrarily for some $P_0\in X_0$ and define
$f(P)$ as $f(P_0)+c(P,P_0)$.) The $1$-cocycle $c$ is a $1$-coboundary
if such an~$f$ can be found satisfying $f(\gm^{-1}P)=f(P)\,|\,\gm$
for all $\gm\in\Gm$.

For a $1$-cocycle $c$ in the model using a chain complex on~$\H$, or
in a model built on a tesselation, the choice of a base point $P_0$
gives a corresponding group cocycle $\ps_\gm = c(\gm^{-1} P_0, P_0)$
in the standard model in \S\ref{ssect-grcoh}.

\sssect{Dimension~$2$}Here the most convenient choice is a projective
resolution of type \Fcoh, based on the tesselation $\tess$ derived
from a fundamental domain~$\fd$, since $\QQ[F_2]$ is free with basis
$(\fd)$. Any $v\in V$ determines a $2$-cochain by $ (\fd)\mapsto v$,
which is automatically a $2$-cocycle since $F_3=\{0\}$. It is a
$2$-coboundary if there is $c\in \map(X_1,V)^\Gm$ such that
\[v =\sum_{e\in \p\fd} c(e) \= \sum_{e\in E} c(e)\,|\,(1-\gm_e)\,.\]
Since the $c(e)\in V$ can be chosen arbitrarily and since the elements
$\gm\in E$ generate $\Gm$, we have
\be \label{H^2}
H^2(\Gm;V) \;\cong\; H^2(\Fpt^\tess;V)
\;\cong\;V_\Gm \;:=\; V \bigm / \textstyle\sum_{\gm\in \Gm}
V|(1-\gm)\,.\ee
The space $V_\Gm$ is called the space of coinvariants.

For general projective resolutions, the isomorphism \eqref{H^2} is
obtained by evaluating a $2$-cocycle $b$ on a representative of the
fundamental class. With the \indlbl{cp}{cap product}\emph{cap
product}
\[\langle\dv,\dv\rangle:H^2(\Gm;V) \otimes H_2(\Gm;\QQ) \lra
H_0(\Gm;V)\;\cong\; V_\Gm\,,\]
we can formulate this as
$b \mapsto \langle [b],[\Gm\backslash\H]\rangle$.
(See, e.g., \S3, Chap.~V of~\cite{Brn}.)
This is a case of \emph{Poincar\'e duality} for $\Gm\backslash \H$,
which holds since $\Gm\backslash \H$ is a rational cohomology
manifold.

In the model using the chain complex on~$\H$, a $2$-cocycle
corresponds to a $\Gm$-equi\-variant map $b:\H^3\lra V$ satisfying
\be \label{coc2}
b(P,Q,R)+b(P,R,S) \= b(P,Q,S)+b(Q,R,S)\,. \ee
Such a cocycle is not necessarily an alternating function of its three
arguments. If we antisymmetrize it by composition with $A_2$ in
\eqref{antisym}, then we get an alternating cocycle in the same
cohomology class.

\sssect{Cocycles associated to Maass forms}In \eqref{pqrdef} we
defined cocycles $r:\Gm \lra \V\om s$, $p:\Gm\lra \E^\om_s$ and
$q:\Gm\lra \W\om s$ associated to $u\in \E_s^\Gm$ in the standard
model of group cohomology. In the model built on the chain complex
on~$\H$, the corresponding cocycles are defined for $P,Q\in\H$ as
follows:\indind{$r_{P,Q}$, $p_{P,Q}$, $q_{P,Q}$}{}
\beqabclbl{pqr-geom}
\begin{align}\label{r-geom}
r_{P,Q}(\z) &\= \int_P^Q[u,R(\z;\dv)^s]\,,
\displaybreak[0]\\
\label{p-geom}
p_{P,Q}(z)&\=\int_P^Q[u,p_s(\dv,z)]\,,
\displaybreak[0]\\
\label{q-geom}
 q_{P,Q}(z)
&\=\int_P^Q[u,q_s(\dv,z)]\,.
\end{align}
\eeqabc
These cocycles describe the linear maps $\bold r$, $\bold p$ and
$\bold q$ of \S\ref{ssect-Maf-coh} in terms of the description of the
cohomology groups with a tesselation. We identify $q_{P,Q}$ with the
element of $\W\om s$ represented by it. Note that the function
$q_{P,Q}$ is not in $\G^\om_s\subset C^2(\uhp)$ (it may jump across
the path from $P$ to~$Q$). The $\Gm$-equivariance follows from that
of the bracket operator and that of the kernel functions. The cocycle
relation~\eqref{c-co} is ensured by the fact that $[u,v]$ is a closed
form if $u$ and $v$ are $\ld_s$-eigenfunctions of~$\Dt$. See
\eqref{d[]}. The same formulas work for models based on a tesselation
$\tess$, provided $P, Q\in X_0^\tess$.



\subsubsection{Algebraic description of cycles and
chains}\label{ssect-ad} This subsection gives some algebraic results
expressing $1$-chains and $1$-cycles in terms of the group
ring.\medskip

Let $R$ denote the \indlbl{gr}{group ring; $\ZZ[\Gm]$}\emph{group
ring} $\ZZ[\Gm]$ of an arbitrary group~$\Gm$ and $R_0\subset R$ the
augmentation ideal, consisting of $\sum_j n_j \, \gm_j\in R$ such
that $\sum_j n_j=0$. For a finite subset $A$ of~$\Gm$ we consider the
map
\[ \pi_A:R^A\to R_0\,,\qquad \x\mapsto \sum_{\al\in A}
(1-\nobreak\al)\,\x(\al)\,.\]
Since $R_0$ is spanned by the elements $1-\gm$ with $\gm\in\Gm$, the
identity $1-\gm\al = (1-\nobreak\al)+(1-\nobreak\gm)\al$ and an
obvious induction show that the image of $\pi_A$ is the kernel of the
natural map from $R$ to $\ZZ[\Gm/\Dt]$, where $\Dt$ is the subgroup
of $\Gm$ generated by~$A$. In particular, $\pi_A$ is surjective if
(and only if) $A$ generates~$\Gm$. As to its kernel, we have:
\begin{lem} Suppose that $A$ generates~$\Gm$ and $\x\in \ker\pi_A$.
Then
\be\label{psiAsum}
 \sum_{\al\in A} \ps_\al \,|\,\x\,(\al) \=0
 \ee
holds for all $\QQ[\Gm]$-modules $V$ and all cocycles
$\ps \in Z^1(\Gm;V)$.
\end{lem}
\begin{proof}
For a coboundary $\ps=db$, we have
$ \sum_{\al\in A} \ps_\al \,|\,\x\,(\al) = -b\,|\, \pi_A\,(\x)$. So
if $\pi_A(\x)=0$, then \eqref{psiAsum} holds for coboundaries. Any
$\QQ[\Gm]$-module is a submodule of an injective
$\ZZ[\Gm]$-module~$I$, for which $H^1(\Gm;I)=\{0\}$. (See \cite{Go},
\S1.4, Th\'eor\`eme 1.2.2.) So any cocycle $\ps\in Z^1(\Gm;V)$ is a
coboundary in $B^1(\Gm;I)$, and \eqref{psiAsum} holds for all
cocycles.
\end{proof}

Now we again take $\Gm$ to be a discrete cocompact subgroup of $G$,
and choose $A=A^{-1}$ to be the system of generators associated to a
Dirichlet fundamental domain~$\fd$ with base point~$P$. Let $\tess$
be the tesselation of type~\Dcoh\ associated to~$\fd$. The edges in
$X_1^\tess$ starting from $P$ are of the form $e_{P,\al^{-1}P}$ with
$\al$ running through the set~$A$. Since every oriented edge in
$X_1^\tess$ is the image under $\Gm$ of one of these, any $1$-chain
$C\in\ZZ[X_1^\tess]$ can be written as
$\sum_{\al\in A} e_{P,\al^{-1}P}\,|\,\x(\al)$ for some $\x\in R^A$.
Then $\p_1C= \sum_{\al\in A}\bigl([P]-[\al^{-1}P]\bigr)\,|\,\x(\al)
=[P]\,|\,\pi_A(\x)$, so $C$ is a $1$-cycle if and only if
$\x\in \ker\pi_A$.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{From cohomology to Maass forms}\label{sect-co-Maf-cc}
In \S\ref{ssect-Maf-coh} we constructed an injective map from
$\E_s^\Gm$ to $ H^1(\Gm;\V\om s)$ for any infinite discrete subgroup
$\Gm$ of~$G$. In this section we prove the bijectivity of this map
when $\Gm$ is cocompact
(Theorem~\ref{thmA}), and give explicit descriptions of the inverse
map $H^1(\Gm;\V\om s)\to \E_s^\Gm$.

The fact that $H^1(\Gm;\V\om s)$ and $\E_s^\Gm$ are isomorphic for
cocompact groups~$\Gm$ follows from the work of Bunke and Olbrich.
\footnote{By Theorem~1.1 in~\cite{BO95} the hyperfunction cohomology
group $H^1(\Gm;\V{-\om}s)$ has finite dimension for torsion-free
cocompact $\Gm$. It is the dual of the analytic cohomology group
$H^1(\Gm;\V\om{1-s})$; this is obtained in Proposition~5.2 with
Poincar\'e duality. Proposition~7.2 in \cite{BO98} gives the same
duality for the distribution cohomology group $H^1(\Gm;\V{-\infty}s)$
and the smooth cohomology group $H^1(\Gm;\V\infty{1-s})$. Corollary
7.3 states the equality $H^i(\Gm;\V{-\infty}s) = H^i(\Gm;\V{-\om}s)$
in all degrees~$i$.

Without the assumption of cocompactness the isomorphism of
$H^1(\Gm;\V{-\infty}s)$ with a space of Maass forms is derived in
\cite{BO98}. The vector bundle $E$ in \S2 is, in our situation, the
constant vector bundle $\CC$ over $G/K \cong\uhp$. The operator $B$
in \S3 corresponds to $\Dt-\ld_s$. The space $\mathcal{E}_Y$ in \S6
corresponds to $C^\infty(\Gm\backslash\uhp)$. The space
$\mathcal{E}_Y(B)_{\mathrm{cusp}}$ at the bottom of p.~71 is the
space $\E^\Gm$ of invariant eigenfunctions if $\Gm$ is compact. If
$\Gm$ has cusps it is the space $\Mf_s^0(\Gm)$ of Maass cusp forms.
Proposition 8.1 in~\cite{BO98} gives the isomorphism between
$\mathcal{E}_Y(B)_{\mathrm{cusp}}$ and $H^1(\Gm;\V{-\infty}s)$. } Our
approach is different and more elementary, and will also form the
basis for the proofs in the non-cocompact case.

The map from Maass forms to cohomology was given in three
versions~$\bold p$, $\bold q$ and $\bold r$ in \S\ref{ssect-Maf-coh}
(defined by \eqref{pqrdef} for the standard model of cohomology and by
\eqref{pqr-geom} for the model based on a tesselation), depending
whether we use the model $\V\om s$, $\W\om s$ or $\E_s^\om$ for the
analytic vectors in the principal series representation. For the
inverse direction we will work with $\W\om s$ and the map~$\bold q$.
We construct an explicit one-sided inverse of~$\bold q$ in
\S\ref{ssect-coMacc}, and give a second description of it in
\S\ref{ssect-coMaavcc}. The injectivity of this inverse map is proved
in \S\ref{ssect-injcc}. Most of the proofs use the description of
cohomology with resolutions based on a tesselation discussed in
\S\ref{ssect-projres}; in Theorem~\ref{thm-cc-upsi} we also give a
formulation in terms of the standard model of group cohomology.


\subsubsection{Construction of a Maass form from a given
cocycle}\label{ssect-coMacc}
 We start with a cocycle $\ps \in Z^1(\Fpt^\tess;\W\om s)$, given in a
 resolution based on a tesselation $\tess$ as described in
 \S\ref{ssect-projres}. This means that for each edge $e\in X_1^\tess$
the boundary germ $\ps(e)\in \W\om s=\G^\om_s/\N^\om$ is given. To
make this concrete, we take representatives $\tilde\ps(e)$ in
$\G^\om_s$ of the $\ps(e)$. This can be done in a $\Gm$-equivariant
way: $\F^\tess_1$ has a finite $\QQ[\Gm]$-basis $B\subset X_1^\tess$,
and we lift each $\ps(b)\in\W\om s$ to $\tilde\ps(b) \in \G^\om_s$
and then extend by $\Gm$-equivariance to get a cochain in
$C^1(\Fpt^\tess;\G^\om_s) = \Hom_{\QQ[\Gm]}(X_1^\tess;\G^\om_s)$.
This is in general not a cocycle, but the cocycle property $d\ps=0$
implies that the values of $d\tilde\ps$ are in $\N^\om =  C^2_c(\H)$.
This defines an $\N^\om$-valued $2$-cocycle which we still denote by
$d\tilde \ps$, although it is now no longer a coboundary.

We now construct a $\ld_s$-eigenfunction $u_\psi$ of $\Dt$ on $\H$.
Recall that the set of singularities $\sing f$ of an element
$f\in\G^\om_s$ is defined as the smallest subset of $\H$ outside of
which $f$ is a $\ld_s$-eigenfunction of the Laplace operator, and is
compact. For compact $x\subset \H$ we denote by
\indlbl{NR}{$N_r(\dv)$}$N_r(x)$ the $r$-neighborhood of $x$ with
respect to the hyperbolic distance. We choose $R$ such that
$\sing\tilde\ps(b)$ is contained in $N_R(b)$ for each~$b$ in the
finite set $B$. Then by $\Gm$-equivariance it follows that
$\sing \tilde\ps(e)$ is contained in the $R$-neighborhood $N_R(e)$
of~$e$ for every edge $e\in X_1^\tess$. For $z\in\H$ we define
\be \label{uCdef}
u_{\tilde\psi}(z) \= \frac1{\pi i}\, \tilde\ps(C)(z)\,,\ee
where $C$ is a cycle in $\ZZ[X_1^\tess]$ going around $z$ once in the
positive direction at a distance greater than~$R$. We claim that this
is independent of the choice of~$C$. Indeed, if $C_1$ is another
$1$-cycle with the same properties as $C$, we can deform $C$ into
$C_1$ in finitely many steps, where in each step we add to or
subtract from $C $ the boundary of a polygon $V\in X_2^\tess$ whose
distance to $z$ is greater than~$R$. The difference between
$\tilde\ps(C)(z)$ and $\tilde\ps(C_1)(z)$ is the sum of contributions
$\tilde\ps(\p V)(z)=d\tilde\ps(V)(z)$. Each function $d\tilde\ps(V)$
is a $\ld_s$-eigenfunction outside of $N_R(V)$ and is compactly
supported, so vanishes identically outside of $N_R(V)$ (because
$N_R(V)$ is simply connected). In particular, each $d\tilde\ps(V)$
vanishes near $z$, so \eqref{uCdef} is the same for $C$ and $C_1$.
(An alternative argument would be to choose a larger $R$ for which
$d\tilde\ps(V)$ vanishes outside the $R$-neighborhood $N_R(V)$ for
every $V\in X_2^\tess$, which is possible by equivariance since
$X_2^\tess/\Gm$ is finite. Then the vanishing of $d\tilde\ps(V)$ near
$z$ is immediate.)


The function $u_{\tilde\psi}$ satisfies
$\Dt u_{\tilde\psi}=\ld_su_{\tilde\psi}$, since by the definition of
$R$ the point $z$ lies outside the singularities of $\tilde\ps(e)$
for every $e$ in $C$. It is also obviously $\Gm$-invariant, since we
can use the cycle $\gm C$ in defining $u_{\tilde\psi}(\gm z)$ and
$\tilde\ps$ is equivariant. Moreover, $u_{\tilde\psi}$ is independent
of the lifting $\tilde\ps$, and can hence be denoted
simply~\indlbl{u-ps}{$u_\ps$}$u_\ps$, because any two choices of
$\tilde\ps$ differ by an equivariant $C_c^2(\H)$-valued on
$X_1^\tess$, so that if we choose the cycle $C$ far enough away from
$z$ the two values of $\tilde\ps(C)$ agree. Finally, $u_\ps$ depends
only on the cohomology class of $\ps$, because if we replace $\ps$ by
another cocycle $\ps_1=\ps+dF$ in the same class, where $F$ is an
equivariant map from $X_0^\tess$ to $\W\om s$, then we can lift $F$
equivariantly to a map $\tilde F: X_0^\tess\to\G^\om_s$ and hence,
choosing $C$ in \eqref{uCdef} suitable for $\ps$ and $\ps_1$, find
$u_\ps(z)-u_{\ps_1}(z)=
u_{\tilde\ps}(z)-u_{\tilde\ps+d\tilde F}(z) = \tilde F(\p C) = 0$.
This completes the construction of the map
$H^1(\Gm;\W\om s)\to\E_s^\Gm$.

We can also use the isomorphism between the various models of
cohomology to write $u_{[\ps]}$ in terms of the standard model. We
first observe that our construction is independent of the tesselation
chosen, since any two tesselations are contained in a common one and
a cycle $C$ that works in~\eqref{uCdef} for a given tesselation also
works for any finer one. If we use a tesselation of type \Dcoh, with
a set $A=A^{-1}\subset\Gm$ of generators of $\Gm$ giving the
transition from the Dirichlet fundamental domain $\fd$ to the
adjacent fundamental domains $\al^{-1}\fd$, then every edge
$e\in X_1^\tess$ can be written uniquely as
$\gm^{-1}e_{\al^{-1}P_0,P_0}$ with $\gm\in\Gm$ and $\al\in A$, where
$P_0\in \mathring \fd$\indlbl{int}{$\mathring \fd$} is the base point
of the tesselation, so we can associate to any group cocycle
$c:\Gm\to\W\om s$ a cocycle $\ps$ on $X_1^\tess$ by setting
$\psi(e)=c(\al)|\gm$. It also follows that $u_\ps$ has the property
stated in Theorem~A that
(up to a constant factor depending on the normalization) it can be
represented on any compact subset of $\H$ by a fixed finite
$\ZZ$-linear combination of functions $\tpoiss s(\ph_\gm)|\gm'$ with
$\gm,\,\gm'\in\Gm$, where $\ph:\Gm\to\V\om s$ represents the
cohomology class.

\smallskip

We now show that the map just constructed is a left inverse
to~$\bold q$. Start with $u\in \E_s^\Gm$. The class $\bold q u$ is
represented by the map $q:X_1^\tess\to\W\om s$ defined by
\[ q(e) (z) \= \int_e \left[ u,q_s(\dv,z)\right]\,.\]
Notice that the element $q(e)$ itself is not in $\G_s^\om$, because it
is singular on~$e$. We choose a lifting of $q$ to a map
$\tilde\ps:X_1^\tess\to\G_s^\om$ by multiplying $q(b)$ for $b\in B$
by a smooth function that is $1$ near $\p\H$ and 0 near $b$ and then
extending equivariantly. Now we apply formula~\eqref{uCdef} with $C$
chosen far enough from $z$ that $\tilde\ps(C)$ and $q(C)$ agree near
$z$, obtaining by Theorem~\ref{thm-psC} the identity
\be\label{u-psi}
 u_{\bold q u}(z) \= u_{\tilde\ps}(z) \= \frac1{\pi
 i}\,\tilde\ps(C)(z)
  \= \frac1{\pi i}\,\int_C \left[ u, q_s(\dv,z) \right] \= u(z)\;.\ee
\smallskip

In summary, we have constructed an explicit map
\indlbl{al-cp}{$\al_s^\om$}$\al_s^\om:[\ps]\mapsto u_\ps$ from
$H^1(\Gm;\W\om s)$ to $\E_s^\Gm$ such that the restriction of the
function $u_\ps$ to any compact subset of~$\H$ is a finite linear
combination of translates $\ps_\gm\,|\,\gm'$ ($\gm,\gm'\in \Gm$), and
 such that $u_{\bold q u}=u$ for $u\in \E^\Gm_s$.


\subsubsection{Construction of a Maass form from a cocycle as an
average}\label{ssect-coMaavcc}
In the previous subsection, we constructed the Maass from $u_\ps$
associated to a $\W\om s$-valued cocycle $\ps$ using the surjective
map $\G^\om_s \lra \W\om s$, where $\G^\om_s$ is a space of functions
defined on the whole of $\H$. (Recall that $\W\om s$ is a space of
boundary germs.) We now give an alternative description in which
$u_\ps$ is represented as the sum of the $\Gm$-translates of a
compactly supported function. This will be used in
\S\ref{ssect-injcc} for the proof of the injectivity of the map
$[\ps]\mapsto u_\ps$.

We choose our tesselation $\tess$ so that there is a fundamental
domain $\fd$ for $\Gm\backslash\H$ consisting of finitely many
elements of $X_2^\tess$. (This can be done by choosing $\tess$ of
type \Fcoh\ or by refining any given tesselation appropriately.) By
deforming the 1-cycle~$C$ used in the definition of $u_\psi$, we can
assume that it bounds a region $D$ consisting of finitely many
$\Gm$-translates of this fundamental domain. Then
\[ \pi i\, u_\ps(z) \= \tilde\ps(C)(z) \= \tilde\ps(\p D)(z) \=
d\tilde\ps(D)(z) \= \sum_{\gm\in \Gm;\, \gm^{-1}\fd \text{ inside }C
} d\tilde\ps(\fd)(\gm z)\,.\]
But $d\tilde \ps(\gm^{-1}\fd)(z) = d \ps(\gm^{-1}\fd)(z) = 0$ for
$\gm^{-1}\fd$ outside $C$, because $\ps$ is a cocycle. Hence
  \be\label{ups-sum} u_\ps(z) \= \frac1{\pi i} \sum_{\gm\in \Gm}
  d\tilde\ps(\fd)(\gm z)\,.\ee

Let us define the \emph{averaging operator} on~$\Gm$ for $f:\H\lra\CC$
as follows:\indind{average; $\av \Gm$}{} \
\be \label{avGdef}
\av \Gm(f) \= f\,|\,\av \Gm \= \sum_{\gm\in \Gm} f\,|\,\gm\,. \ee
(This is not really an average since we do not divide by $|\Gm|$, but
the term is convenient.) If the sum converges absolutely on~$\H$, the
result is a $\Gm$-invariant function. For compactly supported $f$ the
sum $\av \Gm f$ is locally finite, and hence absolutely convergent.
Note that the function $d\tilde\ps(\fd)$ is compactly supported. We
have obtained:
\begin{prop}
Suppose that the $\Gm$-invariant tesselation~$\tess$ contains a
fundamental domain~$\fd$. Then for $\ps \in Z^1(\Fpt^\tess;\W\om s)$
and all lifts $\tilde \ps \in C^1(\Fpt^\tess;\G^\om_s)$ of~$\ps$, we
have
\indlbl{upa}{$u_\ps$}
\[ u_\ps \= \frac1{\pi i}\av \Gm\bigl(d\tilde\ps (\fd)\bigr)\,.\]
\end{prop}

It is remarkable that $\av \Gm\bigl(d\tilde\ps(\fd)\bigr)$ is an
analytic function on~$\H$, whereas
$d\tilde\ps(\fd) \in \N^\om=C_c^2(\H)$ is not
(unless it is zero). We consider this a little more concretely.

Let $f_b$ be representatives of $\ps(b)$ for $b\in B$ and choose a
closed non-selfintersecting curve $\tilde C\subset\H$ (for instance,
a circle near~$\p\H$) such that all the $f_b$ are
$\ld_s$-eigenfunctions of~$\Dt$ on and outside~$\tilde C$. Now choose
$\tilde\psi(b)\in \G^\om_s$ which agrees with $f_b$ outside
$\tilde C$ and has singularities only in the $\e$-neighborhood
$N_\e(\tilde C)$ of~$\tilde C$ for some $\e\ll1$. (For instance, we
can multiply $f_b$ by a smooth function which is 1 outside $\tilde C$
and vanishes on the bounded component of $\H\sm N_\e(\tilde C)$.) We
extend $\tilde\ps$ equivariantly as usual. Then all singularities of
all $\tilde\ps(e)$ are contained in $\Gm N_\e(\tilde C)$, and the
same holds for $d\tilde \ps(\fd)=\tilde\ps(\p\fd)$ and for
$\av \Gm\bigl(d\tilde \ps(\fd)\bigr)$. Moving $\tilde C$ and changing
the $\tilde f_b$ on $\tilde C_\e$ corresponds to adding to
$\tilde \ps$ a cochain with values in $\N^\om=C_c^2(\H)$. This means
for $d\tilde\ps$ that we add an element of
$\sum_{\gm\in \Gm} \N^\om\,|\,(1-\gm)$, which is annihilated
by~$\av \Gm$, so $u_\ps=(\pi i)^{-1}\av\Gm(d\tilde\ps(\fd))$ is
unchanged. Since we can deform $\tilde C$ so that $N_\e(\tilde C)$
avoids the $\Gm$-orbit of any given point in~$\H$, this makes it
clear why $u_\ps$ cannot be singular anywhere.


\subsubsection{Injectivity}\label{ssect-injcc}
It remains to show that the map $\al^\om_s: [\ps]\mapsto u_{[\ps]}$ of
the previous section is injective. This map fits into the commutative
diagram
\be\label{inj-diagr}
\begin{aligned}& \xymatrix{ H^1(\Gm;\W \om s)\; \ar[dd]_{\al^\om_s}
\ar[r]^\dt
& \;H^2(\Gm;\N^\om)\ar[d]_\cong^{\langle\dv,(\fd)\rangle}
&
\\
& \bigl(\N^\om\bigr)_\Gm
\makebox[0cm][l]{$\= \N^\om/\sum_\gm \N^\om|(1-\gm)$} \ar[d]^{\av\Gm}
\\
\E_s(\H)^\Gm\;\ar@{^{(}->}@<-.4ex>[r]^{\subset}
& \;C^2(\H)^\Gm }
\end{aligned}\ee
in which $\dt$ is the connecting homomorphism in the long exact
sequence
\be \label{P-longexact}
\cdots
\lra H^1(\Gm;\G^\om_s)\lra H^1(\Gm;\W\om s)
\stackrel\dt\lra H^2(\Gm;\N^\om)\lra \cdots
\ee
corresponding to the exact sequence~\eqref{NGWex} with $p=\om$ and the
vertical isomorphism is the one given in \eqref{H^2}. We will show
the injectivity of $\av\Gm$ in Proposition~\ref{prop-av-inj} and the
vanishing of $H^1(\Gm;\G_s^\om)$
(and hence injectivity of $\dt$) in Proposition~\ref{prop-H1P0}.
Together with what we have already done this gives a proof of the
following theorem, which is a somewhat more detailed statement of
Theorem~\ref{thmA} in the analytic case.

\begin{thm}\label{thm-cc-upsi}Let $\Gm$ be cocompact. There is an
isomorphism $\al_s^\om:  H^1(\Gm;\V\om s)\rightarrow \E_s^\Gm$, given
by $\ph \mapsto u_{\poiss s \ph}$ as defined in~\eqref{uCdef},
inverting $\bold q$ introduced in~\S\ref{ssect-Maf-coh}.

In the description of cohomology based on a tesselation~$\tess$, the
function $u_{\tpoiss s \ph}$ associated to a given $\V\om s$-valued
cocycle $\ph$ can be given on each compact set in~$\H$ by a finite
linear combination of $\Gm$-translates of $\tpoiss s \ph(b)$ where
$b$ runs through a finite set of edges in~$X_1^\tess$. In the
standard description of cohomology the function $u_{\tpoiss s \ph}$
associated to a cocycle $\ph:\Gm\rightarrow \V \om s$ can on any
compact set in~$\H$ be given as a multiple of a finite sum of
translates $\tpoiss s \ph_\gm|\gm'$
($\gm,\, \gm'\in \Gm$).
\end{thm}

We prove the injectivity of $\av\Gm$ in slightly more generality.
\begin{prop}\label{prop-av-inj}The map
$\av\Gm:C^p_c(\H)_\Gm\to C^p(\H)^\Gm$ is injective for
$p=0,\,1,\dots,\infty$.
\end{prop}
\begin{proof}Let $f\in C^p_c(\H)$ with $\av\Gm(f)=0$. We must show
that $f\in\sum_{\gm\in \Gm}C^p_c(\H)\,|\,(1-\nobreak\gm)$. Choose
$\ch\in C^\infty_c(\H)$ such that $\sum_{\gm\in \Gm}\ch|\gm = 1$
on~$\H$.
(For instance, choose $\Ph\in C^\infty_c(\RR)$ with $C\ne0$, where $C$
is the constant value of the integral
$\int_\H\Ph\bigl(\rho(z,z')\bigr)\, d\mu(z')$, with
$\rho(\dv,\dv)=\cosh\dist(\dv,\dv)$ as in Table~\ref{uhp-disk}
in~\S\ref{sect-efhp}, and set
$\ch(z) = \frac 1C\, \int_\fd \Ph\bigl(\rho(z,z')\bigr)\, d\mu(z')$.)
Then
\[ f\= f - \av \Gm(f)\,\ch \= \sum_{\gm\in \Gm} \bigl( \ch|\gm^{-1}
\cdot f \,-\, \ch \cdot
f|\gm \bigr)
\= \sum_{\gm\in \Gm} \bigl( \ch|\gm^{-1}\cdot
f \bigr)\,|\,(1-\gm)\,. \]
In the last expression we can replace $\Gm$ by the set
$\bigl\{\gm\in \Gm\,:\,\gm^{-1}\supp(\ch) 
\cap \supp(f)\neq\nobreak\emptyset\bigr\}$, which is finite because
both $\ch$ and $f$ have compact support.
\end{proof}

\begin{prop}\label{prop-H1P0}If $\Gm$ is cocompact, then
$H^1(\Gm;\G^\om_s)=\{0\}$.
\end{prop}

The proof of this proposition will occupy the rest of this
subsection.\smallskip

Let $\tess$ be a $\Gm$-invariant tesselation, as in
\S\ref{sect-coh-cc}. We put
\indlbl{Xh0}{$X^H_0$}$X^H_0 = X_0^\tess\cup H$, where $H\subset\p\H$
is the orbit of a fixed point of a hyperbolic element of~$\Gm$. A
given element of $Z^1(\Fpt^\tess;\G^\om_s)$ can be viewed as a
$\Gm$-equivariant function $c:X_0^\tess \times X_0^\tess\to\G^\om_s$
satisfying the cocycle relation \eqref{c-co}. We will extend $c$ to a
function~$c^H$ on $X_0^H\times X_0^H$ with the same properties, with
values in a larger space.

For each $\x\in H$ let $\eta_\x$ be the generator of $\Gm_\x$ for
which $\x=\al(\eta_\x)$, the repelling fixed point of~$\eta_\x$. We
put\indind{Gfsh}{}
\be\label{Gfsh} \G^\fsh_s \= \G^\om_s + \sum_{\x \in H}\av
{\eta_\x}^+\G^\om_s\,,
\ee
where $\av {\eta_\x}^+$ is the one-sided average defined in
\S\ref{ssect-avhyp}. This average maps $\G^\om_s$ into
$\G^\om_s\bigl( \p\H \sm \{\x\}\bigr)$. Thus the elements of
$\G^\fsh_s$ represent germs in the space $\W\fs s$ defined in
\eqref{Wfs-def}. Definition~\ref{singdef} gives $\sing(f)$ for any
$f:\H\rightarrow\CC$ as the smallest set such that $f$ is a
$\ld_s$-eigenfunction on $\H\sm \sing(f)$. For $f\in \G^\fsh_s$, the
set $\sing f$ is not necessarily compact in $\H$. Its closure in
$\proj\CC$ may contain points of~$H$.

\begin{lem}\label{lem-coc-ext}The map $c:X_0\times X_0 \lra \G^\om_s$
corresponding to a cocycle $c\in\map(X_1,\G^\om_s)^\Gm$ can be
extended to a map $c^H:X_0^H\times X_0^H\lra \G^\fsh_s$ satisfying
the conditions~\eqref{c-Gi} and~\eqref{c-co} for a $\Gm$-cocycle.
\end{lem}
\begin{proof}Let $\x = \al(\eta)\in H$, with $\eta=\eta_\x$ as above.
For $P\in X_0$ we set
\be \label{c^H-def}c^H\left(P, \x \right) \=
-c^H\left(\x,P\right)
\= \av \eta^+ c(P,\eta^{-1}P) \in \G^\fsh_s\,.\ee
Since the convergence of $\av \eta^+\bigl( c(P,\eta^{-1}P\bigr)(z)$ is
absolute for each $z\in \H$, we have
\[ \av \eta^+\bigl( c(P,\eta^{-1}P\bigr)(z)
\= \lim_{N\rightarrow\infty} \sum_{n=0}^N
c(\eta^{-n}P,\eta^{-n-1}P)(z)
\= \lim_{N\rightarrow\infty} c(P,\eta^{-N-1}P)(z)\,, \]
where the second equality follows from the cocycle property. For
$P,Q\in X_0$:
\be\label{cPQal}
\begin{aligned}
c^H\bigl( P,\x\bigr) &- c^H\bigl(Q,\x \bigr)
\= \lim_{N\rightarrow\infty}
\sum_{n=0}^N\biggl(c(\eta^{-n}P,\eta^{-n-1}P)
-c(\eta^{-n}Q,\eta^{-n-1}Q ) \biggr) \\
& \= \lim_{N\rightarrow\infty} \biggl( c(Q,P)\,|\,\eta^{N+1} - c(Q,P)
\biggr) \=-c(Q,P)\= c(P,Q)\,.
\end{aligned}\ee
For $\x,\,\x_1\in H$ we define
\be c^H\left( \x,\x_1 \right) \= c^H\left(P, \x_1\right) - c^H\left(
P,\x \right)\,.\ee
In \eqref{cPQal} we see that this does not depend on the choice of
$P\in X_0$. This defines $c^H$ on $X_0^H\times X_0^H$ satisfying the
 cocycle relation.

Let $\x_1=\gm^{-1}\x$ with $\gm\in \Gm$. Then
$\eta_{\x_1} = \gm^{-1}\eta_\x\gm$. The $\Gm$-equivariance follows:
\begin{align*}
c^H \bigl( &\gm^{-1}P,\gm^{-1}\x\bigr) \= c\bigl(\gm^{-1} P,
\gm^{-1}\eta^{-1}\gm\gm^{-1} P\bigr)\,|\,\av {\gm^{-1}\eta\gm}^+\\
& \= c\bigl( \gm^{-1} P, \gm^{-1}\eta^{-1} P \bigr)\,|\,\gm^{-1}\, \av
\eta^+\,\gm \= c\bigl( P, \eta^{-1} P\bigr)
\,|\,\av \eta^+\,\gm \= c^H( P,\x)\,|\,\gm\,.\qedhere
\end{align*}
\end{proof}


The construction of $c^H$ shows that for $P\in X_0$ and
$\x,\x_1\in H$:
\be \overline{\sing c(P,\x)}\cap\p\H \;\subset\;\{\x\}\,,\qquad
\overline{\sing c(\x,\x_1)}\cap\p\H \;\subset\; \{\x,\x_1\} \,. \ee
We have defined $c^H(\dv,\dv)$ as a $\G^\fsh_s$-valued function with
the one-sided averaging operator. Of course, we think of $c^H(P,\x)$
as $c$ evaluated on an infinite path from $P$ to~$\x$.


\begin{lem}\label{lem-chE}Let $c^H$ be as in Lemma~\ref{lem-coc-ext}.
Then $c^H(\x_1,\x_2)\in \E_s$ for all $\x_1,\x_2\in H$.
\end{lem}



\begin{proof} Write $\eta_j$ for $\eta_{\x_j}$ ($j=1,\,2$). We look
for a path $p$ from $\x_1$ to $\x_2$ consisting of three pieces:
\twocolwithpictr{(1) The union
$\bigcup_{n\geq 0} \eta_1^{-n}p_{\eta_1^{-1}P_1,P_1}$, with a chain
$p_{\eta_1^{-1}P_1,P_1}\in\ZZ[X_1]$ from $\eta_1^{-1}P_1$ to $P_1$
for some $P_1\in X_0$.

\noindent (2) a chain $p_{P_1,P_2}\in \ZZ[X_1]$ from $P_1$ to
$P_2\in X_0$.

\noindent
(3) The union $\bigcup_{n\geq 0}\eta_2^{-n}p_{P_2,\eta_2^{-1}P_2}$ for
a chain $p_{P_2,\eta_2^{-1}P_2}$ from $P_2$ to $\eta_2^{-1}P_2$.}{
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\put(-.4,-1.9){$\x_2$}
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\qbezier(-.6,0)(-.3,-.9)(.2,0)
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\put(-1.06,1.06){\line(1,-1){.5}}
\put(-.56,.56){\circle*{.1}}
\put(-.5,.6){$P_1$}
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\put(-.15,-1.2){$P_2$}
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\end{picture}}

As in \S\ref{ssect-coMacc}, there exists large $R>0$ such that
$\sing e $ is contained in the $R$-neigh\-bor\-hood $N_R(e)$ for
every $e\in X_1$. We can choose the path $p$ such that
$N_R(p) = \bigcup_{e\subset p} N_R(e)$ does not intersect any given
compact set~$Z$. Then the singularities of
\[ c^H(p) \= \av {\eta_1}^+ c(\eta_1^{-1}P_1,P_1) + c(p_{P_1,P_2}) +
\av {\eta_2}^+ c(P_2,\eta_2^{-1}P_2) \]
do not meet $Z$. Since $c^H(p) = c^H(\x_1,\x_2)$ does not depend on
the path~$p$, but only on $\x_1$ and $\x_2$, there are no
singularities at all.
\end{proof}


\begin{lem}\label{lem-ch-reg}Let $c^H$ be as in
Lemma~\ref{lem-coc-ext}. Then $c^H(P,\x) \in \G^\om_s$ for all
$P\in X_0$, $\x\in H$.
\end{lem}

\twocolwithpictr{\begin{proof}
We have
$\overline{\sing\left( c^H(P,\x)\right)} \cap \p\H \subset \{\x\}$.
Let $\x_1$ be another point of $H$. Then
$c^H(P,\x)= c^H(P,\x_1) - c^H(\x,\x_1) $, so
$\sing\left( c^H(P,\x)\right) = \sing\left( c^H(P,\x_1)\right) $ by
Lemma~\ref{lem-chE}. Hence
$\overline{\sing\left( c^H(P,\x)\right)} \cap\p\H\break 
\subset \{\x_1\}\cap\{\x\}=\emptyset$. Thus
$\sing \left( c^H(P,\x) \right)$ is a compact subset of~$\H$. Now
apply Lemma~\ref{lem-hyp-cp}, using \eqref{c^H-def}
and~\eqref{av-eta}.
\end{proof}}{
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\qbezier(.5,.5)(.3,0)(0,-.4)
\put(.3,.6){$\sing c^H(P,\x)$}
\qbezier[28](-.33,-1.46)(-.4,.3)(0,.3)
\qbezier[28](0,.3)(0.2,-.8)(-.33,-1.46)
\put(.1,-.8){$\sing c^H(P,\x_1)$}
\end{picture}
}

\begin{lem}We have $c^H(\x_1,\x_2)=0$ for all $\x_1,\,\x_2\in H$.
\end{lem}
\begin{proof}For $P\in X_0$ we have
$c^H(\x_1,\x_2)= c^H(P,\x_2)-c^H(P,\x_1) \in \E_s \cap \G^\om_s$ by
the two preceding lemmas. But $\E_s \cap \G^\om_s=\{0\}$ by virtue of
the splitting~\eqref{F-decomp}.
\end{proof}

\begin{proof}[Proof of Proposition \ref{prop-H1P0}] For a cocycle
$c\in Z^1(\Fpt^\tess;\G^\om_s)$, we have constructed the extension
$c^H$ to $X_0^H\times X_0^H$. For $P\in X_0^\tess$ and $\x\in H$ we
have for all $\gm\in \Gm$
\[ c(\gm^{-1}P,P) \= c^H(\gm^{-1}P,\gm^{-1}\x) + c(\gm^{-1}\x,\x)
+ c^H(\x,P) \= 0 + c^H(P,\x)\,|\,(\gm - 1)\,.\]
Lemma~\ref{lem-ch-reg} shows that $c^H(P,\x) \in \G^\om_s$. Thus
$\gm\mapsto c(\gm^{-1} P,P)$ is a coboundary, and the cohomology
class $[c]\in H^1(\Gm;\G^\om_s)$ is trivial.
\end{proof}









\skipbetweenchapters
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%   Chapter 3   cyclic groups                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\sect{Cohomology of infinite cyclic subgroups of
$\mathrm{PSL}_2(\mathbb{R})$}\label{ch-III}

The general theme of these notes is the relation between
$\Gm$-invariant eigenfunctions and cohomology with values in the
principal series, i.e., between the cohomology groups $H^0(\Gm,\E_s)$
and subspaces of $H^1(\Gm;\E_s)$, where $\Gm$ is a discrete and
cofinite subgroup of $G=\PSL_2(\RR)$.


In this chapter we consider the corresponding question when $\Gm$ is
replaced by an infinite cyclic subgroup
\indlbl{gen}{$\gen\gm$}$\Dt=\gen \gm$ generated by a hyperbolic or
parabolic element $\gm$ of~$G$. This case is of course far easier,
since the structure of $\Dt$ and the geometry of $\Dt\backslash\H$
are much simpler than those of $\Gm$ and $\Gm\backslash\uhp$, so that
we can get very explicit descriptions of the corresponding cohomology
groups. This will give information for the case of real interest,
since the natural morphisms $H^i(\Gm;\E_s) \lra H^i(\Dt;\E_s)$ are
injective for both $i=0$ and $i=1$ and we can therefore identify the
$\Gm$-invariant eigenfunctions and the cohomology groups of $\Gm$
with subspaces of explicit vector spaces. In particular, we show in
\S\ref{ssect-recon} that a $\V\om s$-valued $1$-cocycle corresponding
to an $\Gm$-invariant eigenfunction in $\E_s^\Gm$ can be
reconstructed from its value on a single hyperbolic or parabolic
element of~$\Gm$. And in the Propositions \ref{prop-KH1-sa}
and~\ref{prop-KH1-smp}, and in Theorem~\ref{thm-H1Wow-char} we give a
cohomological characterization of various spaces of eigenfunctions
invariant under a parabolic element of~$\Gm$. These results will be
essential in Chapter~\ref{ch-IV}, where $\Gm$ is a discrete subgroup
with cusps. The results in the present chapter are the technical
heart of these notes.
\medskip


As in \S\ref{ssect-av}, we arrange by conjugation that in the
parabolic case $\Dt=\gen T$, with $T=\matc1101$ leaving fixed
$\infty$, \indlbl{Teta}{$\eta$; $T$, $\Dt$}and in the hyperbolic case
$\Dt=\gen \eta$, with $\eta=\matc t00{1/t}$, $t>1$, leaving fixed
$\al(\eta)=0$ and $\om(\eta)=\infty$. By conjugation the results that
we obtain are valid for general infinite cyclic $\Dt\subset G$. It is
convenient to work in the upper half plane model of~$\H$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Invariants}\label{ssect-inv}
The elements of $\E_s^\Dt$ have a periodicity under $z\mapsto z+1$
(parabolic case) or $z\mapsto tz$ (hyperbolic case). In
\S\ref{ssect-Fe} we discuss the corresponding Fourier expansions, and
in \S\ref{ssect-holf} we show how to associate $\Dt$-invariant
holomorphic functions to elements of $\E_s^\Dt$. In
\S\ref{ssect-recon} we show how we can recover $u$ from the value
$r_\gm$ of a cocycle $r$ representing~$\bold r u$, where
$\bold r: \E_s^\Dt \lra H^1(\Dt;\V\om s)$ is the injection given in
\S\ref{ssect-Maf-coh}.


\subsubsection{Fourier expansion}\label{ssect-Fe}\indind{Fourier
expansion}{ssect-Fe} Recall that Propositions \ref{prop-hyp-inv}
and~\ref{prop-Tinv0} show that $\bigl( \V0s\bigr)^\Dt\nobreak=\{0\}$.
On the other hand, $(\V{-\om}s)^\Dt\cong \E_s^\Dt$ has infinite
dimension. We now consider this space in more detail.

\sssect{Parabolic case} Each $u\in \E_s^T$ has an absolutely
convergent Fourier expansion
\be\label{Fourexp-par} u \= A_0 \, i_{1-s,0}\,+\,B_0 \, i_{s,0} \;+\;
\sum_{n\neq 0} \bigl( A_n \, k_{s,2\pi n} \,+\,B_n \, i_{s,2\pi n}
\bigr)\,. \ee
with $k_{s,\nu}$, $i_{s,\nu}$ as in \eqref{ikdef}.\indind{$A_n$;
$B_n$}{Fourexp-par} For $s=\frac12$ the Fourier term of order zero
must be replaced by $a_0\,i_{s,0}\,+\,b_0\,\ell_{s,0}$ with
$\ell_{s,0}$ defined as in \S\ref{ssect-exef}. The terms with a
factor $B_n$ represent elements of $\W\om s(\RR)$. We will sometimes
write $A_n=A_n(u)$ and $B_n=B_n(u)$.



The modified Bessel functions have the asymptotic behavior
$ K_{s-1/2}(t) \sim e^{-t}\sqrt{\pi/2t} $ and
$I_{s-1/2}(t) \sim e^t/\sqrt{2\pi t} $ as $t\rightarrow\infty$
(\cite{Wat}, \S7.23). This implies the following necessary and
sufficient conditions for the convergence of \eqref{Fourexp-par} in
the upper half plane:
\be\label{ABconv}
A_n\= \oh\bigl( e^{\e|n|}\bigr)\,,\quad B_n\= \oh \bigl(
e^{-|n|/\e}\bigr)\quad\text{ for all }\e>0\,. \ee

For $s\neq \frac12$, we write\indind{$\K_s$; $\I s$;
$\K_s^0$}{Edecomp}
\be \label{Edecomp}
\E_s^T = \K_s\oplus\I s\,, \ee
where elements of $\I s$ have only terms with $i_{s,2\pi n}$ in their
Fourier expansion, and elements of $\K_s$ have only $k_{s,2\pi n}$
and $i_{1-s,0}$. The space $\K_s^0$ inside $\K_s$ is characterized by
the additional condition $B_0=0$. So $\K_s^0 $ is the space of
elements of $\E_s^T$ that have only $k_{s,2\pi n}$, $n\neq 0$, in
their Fourier expansion. This characterization works also to define
$\K_{1/2}^0$, whereas $\K_{1/2}$ is undefined.

The $K$-Bessel function and all its derivatives have exponential decay
at $\infty$, as follows from \S7.23 and~\S3.71 in \cite{Wat}. This
implies that if $u\in \K_s^0$ then $\p_z ^l \p_{\bar z}^m u(z)$ has
exponential decay for all choices of $l,m\in \NN$.


\sssect{Hyperbolic case}Here we use the eigenfunctions $f^R_{s,\al}$
and $f^L_{s,\al}$ in \eqref{frt}. Each $u\in \E_s^\eta$ has an
absolutely convergent Fourier expansion
\be\label{Four-exp-hyp} u \= \sum_\al \bigl( A_\al \, f^R_{s,\al} +
B_\al \, f^L_{s,\al}\bigr)\,, \ee
where $\al$ runs through $\frac{2\pi}{\log t}\ZZ$.
\begin{lem}The coefficients in \eqref{Four-exp-hyp} satisfy
\be\label{ABah} A_\al\,,\; B_\al\;\ll\; e^{-\pi |\al|/2}
\qquad(|\al|\rightarrow\infty)\,.\ee
\end{lem}
\begin{proof}It is convenient not to use the basis
$\bigl\{ f^R_{s,\al}, f^L_{s,\al}\bigr\}$ of the space of functions
transforming according to
$\matc {\sqrt y}00{1/\sqrt y} \mapsto y^{i\al}$, but the basis
$ \bigl\{ f^+_{s,\al}, f^-_{s,\al}\bigr\}$ in (A.17) of~\cite{BLZ}.

We write the Fourier terms as $p_\al f^+_{\al,s} + q_\al f^-_{\al,s}$.
For each $\ph\in (0,\pi)$ the terms in the Fourier expansion of
$u(\rho e^{i\ph})$ are bounded, uniformly in
$\al\in \frac{2\pi}{\log t}\ZZ$. The same holds for the derivatives
with respect to~$\ph$. Then equation (A.18) in~\cite{BLZ} implies
that the $p_\al$ and $q_\al$ are bounded. We express $A_\al$ and
$B_\al$ in $p_\al$ and $q_\al$ by inverting relation
(A.20)
in~\cite{BLZ}, and use of Stirling's formula to obtain~\eqref{ABah}.
\end{proof}



\subsubsection{Holomorphic functions associated to periodic
eigenfunctions}\label{ssect-holf}The growth conditions \eqref{ABconv}
and \eqref{ABah} for the coefficients in the Fourier expansions give
the possibility to encode elements of $\E^\Dt_s$ by a pair of
holomorphic functions or by a holomorphic function and a
hyperfunction.
\sssect{Parabolic case}Let $s\neq \frac12$. For
$u\in \E_s^T$\indind{$\bt(u)$}{btdef}, the formula
\be\label{btdef}
\bt(u)(\z) \= \sum_{n\in \ZZ} B_n \, e^{2\pi in\z}\ee
defines $\bt(u) \in \hol(\CC)^T$, and each element of $\hol(\CC)^T$
occurs in this way. Alternatively, one may use
$\sum_n B_n  \, q^n \in \hol(\CC^\ast)$, with $q=e^{2\pi i\z}$. The
coefficients $A_n$ give rise to\indind{$\al(u)$}{aldef}
\be\label{aldef} \al(u) \= \sum_{n\in \ZZ} A_n \,
\eph_{s,n}^\prj\;\in\; C^{-\om}(\proj\RR)\,,\ee
with $\eph_{s,n}^\prj$ as in~\eqref{phisndef1}. Thus, the bijective
correspondence $u\leftrightarrow \bigl(\al(u),\bt(u)\bigr)$ codes
elements of $\E_s^T$ as pairs consisting of a hyperfunction in
$C^{-\om}(\proj\RR)$ and a holomorphic function in $\hol(\CC)^T$. The
following proposition shows that the function $\bt(u)$ can be related
independently to the $\V\om s$-valued cocycle $r$ associated to $u$
in~\eqref{rdef}.
\begin{prop}\label{prop-bt-par}
Let $s\neq\frac12$, $u\in\E^T_s$, $z_0\in\uhp$.
\begin{enumerate}
\item[i)] With $r_T(\z) = \int_{z_0-1}^{z_0} [u,R(\z;\dv)^s]$, and
with the average $\av T$ in \S\ref{ssect-invavpar} and the gamma
factor $\bb(s)$ in~\eqref{btd}:
\be\label{btui} \bt(u)(\z) \= \frac{- \bb(s)}{\pi i} \av T(r_T)(\z)
\= \frac{- \bb(s)}{\pi i} \int_{iy_0-\infty}^{iy_0+\infty} \bigl[
u,R(\z;\dv)^s\bigr] \,. \ee
\item[ii)] Put $R_s(t;\dv) = \av T\bigl( R(t;\dv)^s\bigr)$. Then
$R_s(t,z)=R_s^1(z-t)$ where $R_s^1\in\E_s^T$ has the Fourier
expansion
\be \label{RsFe}
R_s^1(z) \= \sqrt \pi \frac{\Gf\bigl(s-\frac12\bigr)}{\Gf(s)}
i_{1-s,0}(z) + \frac{2\pi ^s}{\Gf(s)} \sum_{m\neq0} |m|^{s-\frac12}
k_{s,2\pi m}(z)\,.\ee
\item[iii)] The functions $\bt(u)$ and $R_s$ are related by
\be\label{Rse}
\bt(u)(t) \= \frac{-\bb(s)}{\pi i } \int_{z_0-1}^{z_0}
[u,R_s(t;\dv)]\,. \ee
\end{enumerate}
\end{prop}
The integral in \eqref{btui} converges absolutely if $\re s>\frac12$,
and has to be understood in the regularized sense discussed in
\S\ref{ssect-invavpar} if $0<\re s\leq \frac12$.

Before giving the proof of Proposition~\ref{prop-bt-par}, we state the
corresponding result for the hyperbolic case, which will be proved in
\S\ref{ssect-recon}.

\sssect{Hyperbolic case} Elements $u\in\E_s^\eta$ are coded by two
holomorphic functions\indind{$\bt^R(u)$; $\bt^L(u)$}{btlr}
\be\label{btlr}
\begin{aligned}
\bt^R(u)(\z) &\= \sum_\al A_\al \, \z^{i\al}&&\quad\text{ with
}\re\z>0\,,\\
\bt^L(u)(\z) &\= \sum_\al B_\al \,
(-\z)^{i\al}&&\quad\text{ with } \re\z<0\,,
\end{aligned}
\ee
where $\al$ runs through $ \frac{2\pi}{\log t}\ZZ$. The Fourier
coefficients $A_\al$ and $B_\al$ are those in~\eqref{Four-exp-hyp}.
The holomorphic functions $\bt^R (u)$ and $\bt^L (u)$ are invariant
under $\z\mapsto t \z$. We shall prove:
\begin{prop}\label{prop-bt-hyp}Let $u\in \E^\eta_s$, and $z_0=iy_0$,
$y_0>0$. With $r_\eta(\z) = \int_{iy_0/t}^{iy_0} \bigl[
u, R(\z;\dv)^s\bigr]$:
\be \frac{-\bb(s)}{\pi i}\, \av \eta(r_\eta)(\z) \= \begin{cases}
\bt^R(u)(\z)&\text{ for }\re\z>0\,,\\
- \bt^L(u)(\z)&\text{ for }\re\z<0\,.
\end{cases}
\ee
\end{prop}

\begin{proof}[Proof of Proposition \ref{prop-bt-par}] We can assume
that $\re s>\frac32$ since the Fourier expansion \eqref{Fourexp-par}
with fixed coefficients $A_n$ and $B_n$ gives an analytic
continuation of the function $u(z)$ (still belonging to $\E_s^T$),
and hence also of $r_T(\z)$, to all complex values of $s$
(with $\bt(u)\in\hol(\CC)^T$ constant), and since all of the
expressions in \eqref{btui}, \eqref{RsFe} and \eqref{Rse} are
meromorphic in~$s$. For $\re s>\frac 32$ the sum defining the average
$\av T(r_T)$ converges absolutely and by \eqref{[]def} and partial
integration we get for $|\im \z|<y_0$
\begin{align*}
\av T(r_T&)(\z)\= \int_{\im z=y_0} \bigl[u,R(\z;\dv)^{s}\bigr] \=
\int_{\im z=y_0} \frac{-i}2 \bigl\{ u, R(\z;\dv)^{s} \bigr\} \\
&\=
-\frac i2 \int_{-\infty}^\infty \biggl( R(\z;y_0+x)^{s} \,
\frac{\partial
u}{\partial
y}(y_0+x) - u(y_0+x) \, \frac\partial{\partial y} R(\z;y_0+x)^{s}
\biggr)\, dx\,.
\end{align*}
The function $u$ and its derivatives may be unbounded when $z$ varies
in $\uhp$. On a horizontal line however, they stay bounded. We insert
the Fourier expansion of~$u$, and consider the term of order $m$,
which has the form $f_m(y) \, e^{2\pi i mx}$.
\be\label{Wr-int}
\begin{aligned}
-\frac i2 \int_{-\infty}^\infty &e^{2\pi imx} \biggl( f'_m(y_0) \,
R(\z;z)^{s}
- f_m(y_0) \, \frac\partial{\partial y} R(\z;z)^{s}
\bigr|_{y=y_0}\biggr)\, dx
\\
&\= -\frac i2 e^{2\pi im\z} \left( f'_m(y_0)
\, L_m(y_0) - f_m(y_0) \, L_m'(y_0)
\right)\,,
\end{aligned}
\ee
where
\[ L_m(y) \= \int_{-\infty}^\infty e^{2\pi imx} \, R(0;z)^{s}\, dx \=
y^{s} \int_{-\infty}^\infty \frac{e^{2\pi imx}}{(x^2+y^2)^{s}}\,
 dx\,. \]
The last expression is considered in \eqref{Kint}. The contribution of
the term of order $m$ is given by a Wronskian, and can be computed
with use of the definitions in \eqref{ikdef} and (A.4) in \cite{BLZ}.
We find:
\be \label{hFe}
\av T(r_T)(\z) \= - \frac{\pi i\, \Gf(2s)}{2^{2s-1} \, \Gf(s)^2}
\sum_m B_m e^{2\pi im\z} \= -\pi i\, \bb(s)^{-1}\, \bt(u)(\z)\,.\ee
Now equation \eqref{RsFe} is obtained with the summation formula of
Poisson, and finally
\[ \av T(r_T) (\z) \= \sum_n \int_{z_0-n-1}^{z_0-n} [ u, R(\z+n;
\dv)^{s}] \= \int_{z_0-1}^{z_0} [ u, R_s(\z,\dv)]\,.\qedhere\]
\end{proof}



\subsubsection{Reconstruction}\label{ssect-recon}
\indind{reconstruction}{ssect-recon} Proposition~\ref{prop-rinj} has
shown that $u \mapsto r_\gm$ is injective, both for $\gm=T$ and
$\gm=\eta$. To reconstruct $u$ explicitly from $r_\gm$, we first pass
to the image $q_\gm= \bb(s) \tpoiss s r_\gm$ under the transverse
Poisson transform and then reconstruct $u$ from $q_\gm$.

With the base point $z_0\in \uhp$ as in the previous subsection, we
have
\[\av T(q_T)(z) \= \int_{\im z'=z_0} [u,q_s(\dv,z)]\,, \qquad \av
\eta(q_\eta)(z)
\= \int_0^{i\infty} [u,q_s(\dv,z)]\,. \]
The second integral converges absolutely for $\re s>0$, while the
first has to be understood in the regularized sense, under the
assumption $s\neq \frac12$. Both integrals define
$\ld_s$-eigenfunctions of the Laplace operator outside the path of
integration. Thus we have
\indlbl{uDULR}{$u_D$, $u_U$, $U_L$, $u_R$}$u_D$, $u_U$, $u_L$, $u_R$
in $\E_s(X)$, where $X$ in each case is a component of $\uhp$ minus
the path of integration:
\beqabclbl{uhl}
\begin{align}
\label{uhlp}
\int_{\im z'=y_0} [u,q_s(\dv,z)] &\=
\begin{cases}
u_D(z) &\text{ for $z$ below the path}\,,\\
u_U(z) &\text{ for $z$ above the path}\,,
\end{cases}
\\
\label{uhlh}
\int_0^{i\infty} [u,q_s(\dv,z)] &\=
\begin{cases}
u_L(z)&\text{ for $z$ on the left of the path}\,,\\
u_R(z)&\text{ for $z$ on the right of the path}\,.
\end{cases}
\end{align}
\eeqabc
Local deformation of the path of integration shows that these four
functions extend to~$\uhp$, yielding four elements of~$\E_s$.
Theorem~\ref{thm-psC} implies that
\be u_U(z) - u_D(z)\= u_L(z)-u_R(z) \= \pi i\,u(z)\,.\ee

In the \emph{parabolic case}, $u_D$ represents an element of
$\W\om s(\RR)$. We use that the restriction to the boundary $\rho_s$
inverts the transverse Poisson transformation. With \eqref{pqr-rels},
\S\ref{ssect-invavpar} and Proposition~\ref{prop-bt-par}, we have for
$\x\in \RR$:
\[ (\rho_s\av T q_T)(\x) \= \bb(s)\,(\rho_s \av T \tpoiss s r_T)(\x)
\= \bb(s)\, (\av T r_T)(\x) \=
-\pi i\, \bt(u)(\x)\,. \]
Since $i_{s,2\pi n}(z)\sim y^s e^{2\pi inx}$ as $y\downarrow0$, this
implies that
\[ u_D(z) \= -\pi i \sum_n B_n i_{s,2\pi n}(z)\,.\]
This Fourier expansion identifies $-u_D$ as the component of
$\pi i\,u$ in $\I s$ in the direct sum decomposition
$\E_s^T = \K_s \oplus \I s$ of~\eqref{Edecomp}. Then $u_U$ is the
component in $\K_s$. We have obtained:
\begin{prop}\label{prop-q-decomp}Let $s\neq \frac12$. Each function
$u\in \E^T_s$ can be recovered as
\be\label{ureconstr}
 u\=(\pi i)^{-1} u_U- (\pi i)^{-1} u_D\in \K_s \oplus \I s \ee
from the value $q_T\in \W\om s$ of the cocycle $q$ in \eqref{qdef}
representing $\bold q u$, by the regularized integral
in~\eqref{uhlp}.
\end{prop}

\noindent\emph{Remark.} By \eqref{ex1c} the transverse Poisson
transform of the function $R_s(\dv,z')$ occurring in part~ii) of
Proposition~\ref{prop-bt-par} is the resolvent kernel function
\be Q_s(z,z')\= \bb(s)^{-1}\,\bigl( \av T q_s(z,\dv) \bigr)(z')
\ee
on $\{ (z,z')\in \uhp^2\;:\; \im z\neq \im z'\}$. It satisfies
$Q_s(z,z'+\nobreak1)=Q_s(z,z')=Q_s(z',z)$. On $\im z'<\im z$ we use
$\poiss s e^{i\al x} = i_{s,\al}$ to obtain the expansion
\be\label{Qs-exp}
\begin{aligned}
Q_s&(z,z') \= \bb(s)\, \bigl(\poiss s R(z(\dv,z)\bigr)(z')\\
&\= \frac1{s-\frac12} \,i_{1-s,0}(z)\,i_{s,0}(z') +
\frac{2\pi^{s+\frac12}}{\Gf(s+\frac12)} \sum_{m\neq 0}
|m|^{s-\frac12} \,k_{2,s\pi m}(z)\, i_{s,-2\pi m}(z')\,.
\end{aligned}
\ee
 \medskip

In the \emph{hyperbolic case}, $u_L$ represents an element of
$\W\om s(-\infty,0)$, and $u_R$ an element of $\W\om s(0,\infty)$.
Thus $(\pi i)^{-1}u_L$ is given by the part of the Fourier expansion
of $u$ with $f^L_{s,\al}$, and $-(\pi i)^{-1}u_R$ by the part
with~$f^R_{s,\al}$:
\be\label{uLuR} u_L \= \pi i \sum_\al B_\al f^L_{s,\al}\,,\qquad u_R
\=
-\pi i \sum_\al A_\al f^R_{s,\al}\,. \ee
Hence $\rho_s u_R(\x) = -\pi i\, \bt^R(\x)$ for $\x>0$, and
$\rho_s u_L(\x) = \pi i \, \bt^L(\x)$ for $\x<0$. This proves
Proposition~\ref{prop-bt-hyp}. Furthermore, we have obtained:
\begin{prop}Let $u\in \E^\eta_s$, and let $q\in Z^1(\gen\eta,\W\om s)$
be a cocycle representing $\bold q u$. The integral in \eqref{uhlh}
reconstructs $u$ from $q_\eta$ as
$u=(\pi i)^{-1} \left( u_L-u_R\right)$, with the two terms
corresponding as in~\eqref{uLuR} to the $f^L$ and $f^R$ terms in the
Fourier expansion~\eqref{Four-exp-hyp}.
\end{prop}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Coinvariants}\label{sect-coinv}

In this section the subject of study is
$\V\om s \bigr/  \bigl(\V\om s\,|\,(1-\nobreak\gm) \bigr)$, for
$\gm=\eta$ and $\gm=T$, as before. The parabolic case $\gm=T$ is more
complicated than the hyperbolic case $\gm=\eta$. The parabolic case
will lead us to consider several $G$-modules between $\V\om s$ and
$\V\fs s$, like $\V\sa s$ and $\V\wdg s$ mentioned in the
introduction.

The main theme in this section is the correspondence between various
spaces of $T$-invariant eigenfunctions and cohomology groups. The
main results are Proposition \ref{prop-KH1-sa},
Proposition~\ref{prop-KH1-smp} and Theorem~\ref{thm-H1Wow-char}. We
will use these and other results from this section in
Chapter~\ref{ch-IV}, where we study the cohomological
characterization of various spaces of $\Gm$-invariant eigenfunctions
for discrete subgroups $\Gm\subset G$ with cusps.



\subsubsection{The first cohomology group and averaging
operators}\label{ssect-H1Av}
Let $\gm=T$ or $\eta$ and $\Dt=\gen\gm$ as in the previous section. We
have $H^1(\Dt;\V\om s) \cong (\V\om s)_\Dt 
\cong \V\om s/\bigl(\V\om s\,|\,(1-\nobreak\gm)\bigr)$, by
associating to $v\in \V\om s$ the cocycle $\ps$ with $\ps_\gm=v$.
(This can be seen as a special case of Poincar\'e duality, since
$V_\Dt=H_0(\Dt;V)$ and the classifying space of $\Dt$ is a circle.)

To apply the averaging operator $\av \gm $ defined in
\S\ref{ssect-av}, we assume in the parabolic case that
$s\neq\frac12$. Since $v\,|\,(1-\nobreak\gm)\,|\,\av \gm^\pm = v$,
the space $\V\om s\,|\,(1-\nobreak\gm)$ is contained in the kernel of
$\av \gm = \av\gm^+-\av\gm^-:\V\om s \lra \bigl(\V\fs s\bigr)^\gm$.
So $\av \gm$ induces a linear map
$H^1(\Dt;\V\om s) \lra \bigl(\V\fs s\bigr)^\Dt$.

The hyperbolic case is easy to treat. Here the image of $\av\gm$ is
contained
$(\V\om s[0,\infty])^\Dt\allowbreak\cong\V\om s(-\infty,0)^\Dt
\oplus\V\om s(0,\infty)^\Dt$
(cf.~\eqref{V[]}), and we have:
\begin{prop}\label{prop-hc}Let $\Dt=\gen\eta$ with $\eta$ hyperbolic.
Then the map $\av \eta:H^1(\Dt;\V\om s)\lra \V\om s[0,\infty]^\Dt$ is
injective, and the natural map
$H^1(\Dt;\V\om s) \lra H^1(\Dt;\V\fs s)$ is zero.
\end{prop}
\begin{proof}The first statement is equivalent to
Corollary~\ref{cor-avh}. For $v\in \V\om s$ we set
$h=\av \eta(v)\in\V{}s[0,\infty]^\Dt$ and define $f\in \V\fs s$ by
$f(x)= -\frac{\log |x|}{\log t}\, h(x)$. Since $h|\eta=h$, the
function~$f$ satisfies $f|\,(1-\nobreak\eta)=h$.
\end{proof}

The parabolic case is more complicated. In that case the map
$\av T : (\V\om s)_\Dt \lra (\V\om s[\infty])^\Dt$ is no longer
injective.



\subsubsection{Invariant eigenfunctions on subsets of the upper
half-plane} In the main theorems of these notes, we give isomorphisms
between on the one hand spaces of Maass forms and larger spaces of
invariant eigenfunctions, and on the other hand cohomology groups. In
this section we study similar relations, but now between spaces of
$T$-invariant eigenfunctions and cohomology groups for $\gen T$. This
subsection serves to define such spaces of $T$-invariant
eigenfunctions.\smallskip

\begin{defn}We put\indind{$\E_s^\uparrow,\;\E_s^\downarrow$}{Eupdown}
\begin{equation}\label{Eupdown}
 \E_s^\uparrow \= \limind Y \E_s\bigl(\{z\in \uhp\;:\; \im
 z>Y\}\bigr)\,,\quad \E_s^\downarrow \= \limind \e \E_s\bigl(\{z\in
\uhp\;:\; 0<\im z<\e\}\bigr)\,.
\end{equation}
\end{defn}
Thus, elements of $\E_s^\uparrow$ may be viewed as eigenfunctions
defined on some half-plane $\im z>Y$, where $Y$ may depend on the
function. Similarly, elements of $\E_s^\downarrow$ are defined on
some strip $0<\im z<\e$. Representatives of elements of
$(\E^\uparrow_s)^T$ and $(\E_s^\downarrow)^T$ have Fourier expansions
of the form indicated in~\eqref{Fourexp-par}
(modified as indicated there if $s=\frac12$)
converging on some half-plane or strip. The Fourier coefficients
satisfy weaker growth conditions than those indicated
in~\eqref{ABconv}, namely
\be
\begin{aligned}
u\in(\E_s^\uparrow)^T &\Leftrightarrow A_n=
\oh\bigl(e^{\e|n|}\bigr)\text{ for some }\e>0\text{ and } B_n =
\oh\bigl(e^{-|n|/\e}\bigr)\text{ for all }\e>0\,,\\
u\in (\E_s^\downarrow)^T&\Leftrightarrow A_n=
\oh\bigl(e^{\e|n|}\bigr)\text{ for all }\e>0\text{ and } B_n =
\oh\bigl(e^{-|n|/\e}\bigr)\text{ for some }\e>0\,.
\end{aligned}
\ee

\begin{defn}
For $s\neq \frac12$, we define
\indlbl{Ku0}{$\K_s^\uparrow,\;
(\K^0_s)^\uparrow$}$\K_s^\uparrow = \K_s \cap \E^\uparrow_s$ as the
subspace of elements of $(\E_s^\uparrow)^T$ represented by functions
with only terms $k_{s,2\pi n}$, $n\neq0$, and $i_{1-s,0}$ in their
Fourier expansion. For all $s$ we define $(K_s^0)^\uparrow$ as the
subspace of $(\E_s^\uparrow)^T$ of elements represented by a Fourier
expansion containing only terms $k_{s,2\pi n}$, $n\neq 0$, and
$\I s^\downarrow$\indlbl{Id}{$\I s^\downarrow$} as the subspace of
elements of $(\E_s^\downarrow)^T$ with representatives containing
only terms $i_{s,2\pi n}$ in their Fourier expansion.
\end{defn}

The cocycles $ r$, $ p$ and $ q$ on $\gen T$ in \eqref{pqrdef} make
sense for $u\in (\E_s^\uparrow)^T$, provided we take $\im z_0$
sufficiently large. The proof of Proposition~\ref{prop-bt-par} can be
extended to give:
\begin{lem}\label{lem-uupbt}\indind{$\bold r $;
$\bt(u)$}{lem-uupbt}Let $s\neq \frac12$. Let
$u\in (\E_s^\uparrow)^T$. The cohomology class $\bold r u$ is
represented by the cocycle $r$ determined by
$r_T(\z) = \int_{z_0-1}^{z_0} [u,R(\z;\dv)^s]$ for $\im z_0$
sufficiently large. The average $\av T(r_T)\in \V\om s[\infty]^T$ is
represented by the holomorphic function
$-\pi i\, \bb(s)^{-1}\,\bt(u)$, where $\bt(u)\in \hol(\CC)^T$ is
defined by an expansion similar to \eqref{btdef} with $B_n$ the
coefficient of $i_{s,2\pi n}$ in the Fourier expansion of~$u$.
\end{lem}
We note that although $u\in (\E^\uparrow_s)^T$ is represented by a
function on some half-plane $\im z>Y$, the corresponding series
$\sum_n B_n i_{s,2\pi n}$ converges on all of~$\uhp$, and hence
$\bt(u)$ is holomorphic on~$\CC$.


The average $\av T (r_T)$ in Lemma~\ref{lem-uupbt} is defined on a
neighborhood of $\RR$ in~$\CC$. The average $\av T(q_T)$ makes sense
on two regions in~$\uhp$. We now consider the consequences of this
fact.

Let $h$ be a function on~$\uhp$ representing an element of~$\W\om s$.
Such an element is in $\E_s(\uhp\sm C)$ where $C\subset\uhp$ is
compact. The averaging operator defines two $\ld_s$-eigenfunctions of
the Laplace operator, on regions $\im z>Y$ and $0<\im z<\e$, where
$\e>0$ and $Y>0$ are such that the regions
$\{z\in \uhp\;:\; y\leq \e\}$ and $\{z\in \uhp\;:\; y\geq Y\}$ are
both contained in the domain of~$h$. We obtain:
\be\label{av-up-down}
\begin{aligned}
\av T^\uparrow(h)&\text{ represented by }\av T(h) \text{ on a region }
\im z\geq Y\,,\\
\av T^\downarrow (h) &\text{ represented by }\av T(h)\text{ on a
region }0<\im z<\e\,.
\end{aligned}
\ee
\begin{center}
\setlength\unitlength{1.2cm}
\hspace*{\fill}
\begin{picture}(5,2)
\put(0,0){\line(1,0){5}}
\put(5.05,-.15){$\RR$}
\put(4,2){$h$}
\put(2.8,.8){$C$}
\thicklines
\qbezier(2,.4)(3,.2)(3.8,.4)
\qbezier(3.8,.4)(3.4,1.5)(3,1.6)
\qbezier(3,1.6)(2.7,1.4)(2.2,1)
\qbezier(2.2,1)(2.1,.6)(2,.4)
\end{picture}
\hspace{\fill}
\begin{picture}(5,3)
\put(0,0){\line(1,0){5}}
\put(5.05,-.15){$\RR$}
\put(2.8,.8){$C$}
\put(.7,.05){$\scriptstyle \av T^\downarrow(h) $}
\put(.7,2.2){$\scriptstyle \av T^\uparrow(h)$}
\qbezier(2,.4)(3,.2)(3.8,.4)
\qbezier(3.8,.4)(3.4,1.5)(3,1.6)
\qbezier(3,1.6)(2.7,1.4)(2.2,1)
\qbezier(2.2,1)(2.1,.6)(2,.4)
\thicklines
\put(0,.27){\line(1,0){5}}
\put(4.5,.37){$\scriptstyle \im z=\e$}
\put(0,1.6){\line(1,0){5}}
\put(4.5,1.7){$\scriptstyle \im z=Y$}
\end{picture}
\hspace*{\fill}
\end{center}


\begin{lem}\label{lem-ABdecomp}\indlbl{avTud}{$\av T(\dv)^\downarrow$,
$\av T(\dv)^\uparrow$}\begin{enumerate}
\item[i)] If $s\neq \frac12$, then for all boundary forms
$h\in \W\om s$
\[ \av T^\downarrow(h) \in \I s^\downarrow\,,\qquad \av
T^\uparrow(h)\in \K_s^\uparrow\,.\]
\item[ii)] If the boundary form $h\in \W\om s$ satisfies
$(\rho_s h)^\prj(\infty)=0$ then for all $s$ with $0<\re s<1$
\[ \av T^\downarrow (h)\in \I s^\downarrow\,,\qquad \av
T^\uparrow(h)\in \bigl(\K_s^0\bigr)^\uparrow\,.\]
\end{enumerate}
\end{lem}
\begin{proof}Using the restriction $\rho_s:\W\om s\lra \V\om s$, we
put $\ph=\rho_s h \in \V\om s$. We follow the reasoning in the proof
of Proposition~\ref{prop-bt-par} with $u$ replaced by
$\av T^\downarrow(h)$, keeping the integral on a line with $y_0<\e$.
Thus, we get a holomorphic $1$-periodic function on $|\im \z|<\e$.
Applying $\tpoiss s$, we get a series expansion for
$\av T^\downarrow( h)$ in the eigenfunctions $i_{s,2\pi  n}$.

Next we turn to $\av T^\uparrow(h)$. There is no corresponding
``$\av T^\uparrow(\ph)$''. Let first $s\neq\frac12$.
Lemma~\ref{lem-avWas} implies that
$\av T(h)(z)= C y^{1-s} + \oh(y^{- s})$ as $y\rightarrow\infty$. This
shows that the Fourier expansion of this periodic function consists
of multiples of $i_{1-s,0}$ and $k_{s,2\pi n}$, $n\neq 0$. If
$(\rho_s h)^\prj(\infty)=0$, the same lemma gives an estimate
$\oh(y^{- s})$, which shows that we have only terms $k_{s,2\pi n}$,
$n\neq0$.
\end{proof}


\begin{prop}\label{prop-udecomp-up}For $u\in (\E_s^\uparrow)^T$ put
$h(z) = \int_{z_0-1}^{z_0} [u,q_s(\dv,z)]$. Then
$\av T^\downarrow(h) \in \I s$, and $u$ is reconstructed from $h$ by
\be\label{uh-reconstr} u(z) \= (\pi i)^{-1} \av T^\uparrow(h)(z)
- (\pi i)^{-1}\av T^\downarrow
(h)(z)\qquad(\text{$\im z$ sufficiently large})\,, \ee
giving the decomposition
$(\E_s^\uparrow)^T =  \K^\uparrow_s\oplus \I s$, which
generalizes~\eqref{ureconstr}.
\end{prop}
\begin{proof}The definition of $h$ generalizes the cocycle $q$
in~\eqref{qdef}. We have to take $z_0$ in the domain of a
representative of~$u$. The function $h$ is in $\W\om s$ by the
properties of the kernel function~$q_s$. Following the reasoning in
\S\ref{ssect-recon} we get the decomposition \eqref{ureconstr}, and
obtain also that $\av T ^\downarrow(h)(z)$ doe depend on $z_0$ with
$\im z_0>\im z$. This implies that $\av T^\downarrow (h) \in \I s$.
\end{proof}




\subsubsection{Smooth semi-analytic vectors}It will turn out, in
\eqref{Pal-sa} and \eqref{Hpa}, that the period functions $\ps$
attached to modular Maass cusp forms in the introduction are elements
of the space
\indlbl{Vsa}{$\V sa s$}$\V\sa s $ defined in~\eqref{Vfs-cond}, which
is equal to $\V \fs s \cap \V \infty s$. We call it the space of
\indlbl{smsav}{smooth semi-analytic vectors}\emph{smooth semi-analytic
vectors} in $\V{}s$.

For $\W{}{}$ we use the analogous notational convention
$\W{\fs\!,\,\mathrm{cond}}s$ as for $\V{}{}$ in~\eqref{Vfs-cond} and
below~\eqref{V[]}, where ``cond'' denotes a condition imposed at the
singularities.

The space \indlbl{Wsa}{$\W\sa s$}$\W\sa s$ consists of the elements of
$\W\fs s$ with representatives
$f(z) = \left( \frac y{|z+i|^2}\right)^sa^\prj(z)$ with $a^\prj$ real
analytic on an open set $\Om\subset \proj\CC$ with $\proj\RR\sm \Om$
finite, such that $\rho_s: \x\mapsto a^\prj (\x)$ on
$\proj\RR\cap \Om$ extends as an element of $\V\infty s$.

The $G$-modules $\V\sa s$ and $\W\sa s$ are isomorphic with the
inverse isomorphisms\indind{$\rho_s$; $\tpoiss s$}{rTW}
\be \label{rTW}
 \rho_s: \W\sa s \lra \V\sa s\,,\qquad \tpoiss s : \V\sa s \lra \W\sa
 s\,. \ee

We can produce elements of $\V\sa s$ by the following integral,
discussed in \cite{LZ}, Chapter~2, \S2:
\begin{prop}\label{prop-Vominf}Let $u\in (\K_s^0)^\uparrow$. Then
\be\label{Vominfint}
f_{z_0}(\z) \= \int_{z_0}^\infty \left[ u,R(\z;\dv)^s \right]\,,\ee
with $\re z_0$ sufficiently large, defines
$f_{z_0}\in \V{\om,\infty}s[\infty]$, independent of the path of
integration from $z_0$ to~$\infty$, provided $\infty$ is approached
along a vertical line.

As $z_0$ tends to~$\infty$ along a vertical half-line, then $f_{z_0}$
tends to~$0$ in the topology of $\V\infty s$, defined by the
seminorms $\|\dv\|_n$ in \eqref{seminorm} for all $n\in \NN$.
\end{prop}
The second part of this proposition is one of the few places in these
notes where we mention the topology of principal series spaces. We
will use this part in the proof of Theorem~\ref{thm-Ppcp}.
\begin{proof} In part~I, the first statement has been proved with the
proposition in \S2 of Chap.~II and use of $L$-functions. Here we also
consider the limit in the topology of~$\V\infty s$.

It suffices to consider $z_0=iy_0$ with $y_0>1$. We work in the
projective model. {}From \eqref{R-othermodels} and \eqref{[]def} we
conclude for $\z\in \RR$:
\be \label{R-int}
f_{iy_0}(\z) \= \int_{y_0}^\infty y^s\, \biggl(
\frac{\z^2+1}{\z^2+y^2} \biggr)^s\, \biggl( i\, u_z(iy) + \frac s{2y}
u(iy)
\frac{\z-iy}{\z+iy}\biggr)\, dy\,. \ee
The exponential decay of $u$ and $u_z$ implies convergence. The
coefficients in the expansion in $\z-\z_0$ of
$\biggl( \frac{\z^2+1}{\z^2+y^2} \biggr)^s$ and of
$y^{-1} \biggl( \frac{\z^2+1}{\z^2+y^2} \biggr)^s \frac{\z-iy}{\z+iy}$
at a given $\z_0\in \RR$ are bounded functions of~$y$. Hence after
integration we get a converging power series representing
$f_{iy_0}(\z)$ on a neighborhood of~$\z_0$. So
$f_{iy_0}\in \V\om s[\infty]$.

On a neighborhood of $\z=\infty$ we use the variable $\x=\frac1\z$:
\be f_{iy_0}(1/\x) \= \int_{y_0}^\infty y^s \, \biggl(
\frac{1+\x^2}{1+y^2\x^2} \biggr)^s \biggl( i \,u_z(iy) + \frac s{2y}
\, u(iy) \, \frac{1-iy\x}{1+iy\x} \biggr)\, dy\,. \ee
This converges as well. The coefficient of $\x^n$ in the expansion of
the integrand at~$\x=0$ contains a term $y^{s+n} u(iy)$. So
analyticity at~$\infty$ seems out of the question. Differentiation
with respect to~$\x$ can be carried out inside the integral. This
shows that $f_{iy_0}\in \V\infty s$.

Next we estimate the supremum norm
$\| f_{iy_0}\,|\,\bold W^n\|_\infty$ on $\RR$ for all $n\in \NN$,
where the differential operator $\bold W$ is given by
$ (1+\nobreak \z^2)\p_\z $ on~$\RR$, and by $-(1+\nobreak \x^2)\p_\x$
on a neighborhood of $\z=\infty$. {}From \eqref{R-int} we check that
$\bigl( (1+\nobreak\z^2)\p_\z\bigr)^n f_{iy_0}(\z)$ is a linear
combination, with coefficients depending only on $s$ and $n$, of
integrals
\[ \int_{y_0}^\infty f_j(y,\z)\, U_{\!j}(y)\; dy\,,\]
where
\begin{align*}
f_j(y,\z) &\= y^{s+a_j} \, \biggl( \frac{\z^2+1}{\z^2+y^2} \biggr)^s\,
(\z-i)^{b_j}(\z+i)^{c_j}(\z-iy)^{-d_j}(\z+iy)^{-e_j}\,,\\
U_{\!j}(y)&\= u_z(iy)\,,\; a_j\=0\,,\qquad\text{ or } \quad
U_{\!j}(y)\= u(iy)\,,\; a_j\=-1\,,\\
b_j,\, c_j&\;\in [0,n]\,,\quad -1\leq d_j\leq n\,,\quad 0\leq e_j \leq
n+1\,,\\
b_j+c_j&\=d_j+e_j+n\,.
\end{align*}
For $\z$ in a bounded interval $[-A,A]$ and $y\geq y_0$ we have
$ f_j(y,\z) \ll_{s,n} A^n y^{ s}$. With the exponential decay of $u$
and its derivatives we obtain on $[-A,A]$:
\be\label{Re} \bigl( (1+\z^2)\p_\z\bigr)^n f_{iy_0}(\z) \;\ll_{s,n,\e}
A^n y_0^{ s} e^{-\e y_0}\,, \ee
with $\e \in (0,2\pi)$.

For $|\x|\leq A^{-1}<1$, we have a similar linear combination of
finitely many integrals
$\int_{y_0}^\infty g_j(y,\x)\, U_{\!j}(y)\;dy$ with $U_{\!j}$ as
before and
\begin{align*}
g_j(y,\x) &\= y^{s+a_j}\,\biggl( \frac{1+\x^2}{1+y^2\x^2} \biggr)^s
(1-i\x)^{b_j}(1+i\x)^{c_j}(1-iy\x)^{-d_j}(1+iy\x)^{-e_j}\,,\\
a_j,\, d_j&\geq -1\,,\quad b_j,\, c_j,\, e_j\in \NN\,,\quad
d_j+e_j\geq 0\,,\quad a_j \leq n\,.
\end{align*}
Now we have $g_j(y,\x) \ll_{s,n} y^{ s+a_j} \, (1+\nobreak
y^2\x^2)^{-(d_j+e_j)/2} \ll y^{n+ s}$. This leads to
\be \bigl(-(1+\x^2)\p_\x\bigr)^n f_{iy_0}(1/\x) \;\ll_{s,n}\; y_0^{n+
s} e^{-\e y_0}\,.\ee
This estimate and \eqref{Re} show that
$\lim_{y_0\rightarrow\infty} \bigl\| f_{iy_0}\,|\,\bold
W^n\bigr\|_\infty = 0$ for each~$n\in \NN$.
\end{proof}



Applying the transverse Poisson transformation $\tpoiss s$ to
$f_{z_0}$ in Proposition \ref{prop-Vominf} and multiplying by the
gamma factor $ \bb(s)$, we obtain with use of~\eqref{ex1c}
\be \label{Wominfint}
h_{z_0} (z) \= \int_{z_0}^\infty [u,q_s(\dv,z)]\,,\ee
representing an element of $\W{\om,\,\infty}s$, also called $h_{z_0}$.



Let $q$ be a cocycle as in \eqref{qdef}, representing $\bold q u$,
$u\in \K_s^0$. We would like to write $q_T = h_{z_0-1} - h_{z_0}$. To
do that, we need the following extension of Theorem~\ref{thm-psC}:
\begin{prop}\label{proppsCi}
Suppose that $C$ is a piecewise smooth positively oriented simple
closed curve in $\uhp \cup \{\infty\}$. Suppose that near $\infty$
the curve $C$ consists of geodesic half-lines. For each
$u\in \K_s^0$:
\[ \frac1{\pi i} \int_C \left[ u,q_s(u_U,z)
\right] \=
\begin{cases} u(z)&\text{ if $z$ is inside $C$}\,,\\
0&\text{ if $z$ is outside $C$}\,.
\end{cases}
\]
\end{prop}
\begin{proof}
Approximate $C$ by $C_a$, where near $\infty$ we have replaced the
part of $C$ in the region $y\geq a$ by a curve $l_a$ along
$\left\{ x+ia\;:\; x\in \RR\right\}$. Theorem~\ref{thm-psC} can be
applied to $C_a$. The integral
$\int_{l_a} \left[ u,q_s(\dv,z)\right]$ tends to zero as
$a\rightarrow\infty$ for each $z\in \uhp$, by the exponential decay
of $u$ and its derivatives. The same holds for the difference of the
integrals over $C$ and $C_a$. The results follows.
\end{proof}
We apply Proposition~\ref{proppsCi} to the curve $C$ that consists of
geodesic paths from $\infty$ to $z_0-1$, to $z_0$, and then back
to~$\infty$. For $z$ outside~$C$:
\[ \int_{z_0-1}^{z_0} [u,q_s(\dv,z)] \= h_{z_0-1}(z) - h_{z_0}(z)\,.\]
So $q_T=h_{z_0-1} - h_{z_0}$ in $\W\fs s$. Application of restriction
map $\rho_s:\W\sa s\lra \V\sa s$ gives for the cocycle~$r$, with
$f_{z_0}\in \V\sa s$ as in~\eqref{Vominfint}:
\be\label{rhh}
r_T \= f_{z_0-1} - f_{z_0}\,.\ee
Thus we have, with $\Dt=\gen T$:
\be
\begin{aligned}
\bold r \K_s^0 &\;\subset\; \ker\bigl( H^1(\Dt;\V\om s) \lra
H^1(\Dt;\V\sa s)\bigr)\,,\\
\bold q \K_s^0 &\;\subset\; \ker\bigl( H^1(\Dt;\W\om s) \lra
H^1(\Dt;\W\sa s)\bigr)\,.
\end{aligned}
\ee

\noindent\emph{Notation. } We will often deal with such kernels of
natural maps between cohomology groups. For $\Dt$-modules
$V\subset W$ we put
\be\label{mcd} H^1(\Dt;V,W) \= \ker \bigl( H^1(\Dt;V) \longrightarrow
H^1(\Dt;W)\bigr)\,.
\ee
For $\Dt=\gen T$ one may view this as a mixed parabolic cohomology
group, as we will discuss in Definition~\ref{def-pcG}. We
reformulate:
\be\label{iminf}
\bold r \K_s^0 \;\subset\; H^1(\Dt;\V\om s,\V\sa s)\,,\qquad \bold q
\K_s^0 \;\subset\; H^1(\Dt;\W\om s,\W\sa s)\,.
\ee


\begin{prop}\label{prop-kerav-inf} The following statements are
equivalent for $\ph\in \V\om s$:
\begin{enumerate}
\item[a)] $\ph^\prj(\infty)=0$ and $\av T^+(\ph) = \av T^-(\ph)$.
\item[b)] There exists $h\in \V\sa s$ such that $h\,|\,(1-T) =\ph $
in~$\V\om s$.
\end{enumerate}
If a) and~b) hold, then there is only one $h\in \V\infty s$ as in~b),
namely $h=\av T^+(\ph)=\av T^-(\ph) $.
\end{prop}
\begin{proof}Suppose that a) is satisfied. Since $\ph^\prj(\infty)=0$
the averaging operators $\av T^+$ and $\av T^-$ converge absolutely
on~$\ph$, without regularization and without the assumption
$s\neq\frac12$. In \eqref{Cmast} the constant $C_0$ vanishes. Hence
the expansions in \eqref{asympt} start at $m=0$. {}From
$\av T^+ (\ph) = \av T^-(\ph)$ we conclude that the averages define
an element $h\in \V{\om,\,\infty} s[\infty]$, which satisfies~b).

Suppose that b) is satisfied. This implies that $\ph^\prj(\infty)=0$.
Hence the averages converge without regularization. If
$s\neq \frac12$ we immediately obtain a)
from Proposition~\ref{prop-kerav-(1)} below. In a proof valid for all
$s$ with $0<\re s<1$, we note that $\av T^+(\ph)$ converges without
regularization and satisfies $\av T^+(\ph)(x) = \oh(|x|^{-2s})$ as
$x\rightarrow\infty$. Since $h$ has the same behavior, and since
$h-\av T^+(\ph)$ is periodic we conclude that $h=\av T^+(\ph)$.
{}From the behavior as $x\downarrow-\infty$ we obtain
$h=\av T^-(\ph)$.
\end{proof}

\begin{cor}\label{cor-avT0}The kernel of
$\av T : \V  \om s\rightarrow \V  \fs s$ contains the space of
$\ph \in \V\om s$ that satisfy the equivalent conditions a) and~b)
in Proposition~\ref{prop-kerav-inf}, but is larger than this space.
\end{cor}
\begin{proof}The first statement follows directly from
Proposition~\ref{prop-kerav-inf}. For the second statement, consider
$u\in \K^0_s$, $u\neq 0$, and put
$\ph(\z) = \int_{z_0-1}^{z_0} [u,R(\z;\dv)^s]$ for some
$z_0\in \uhp$. Then $\ph \in \V\om s$, and it satisfies condition~b)
in Proposition~\ref{prop-kerav-inf} (by
Proposition~\ref{prop-Vominf}). Since $\ph=r_T$ as defined in
\eqref{pqrdef}, the injectivity of $\bold r$ in
Proposition~\ref{prop-rinj} shows that the class $\bold r u$ in
$H^1(\Gm;\V\om s)$ represented by $r$ is non-zero. So
$\ph \not\in \V\om s\,|\,(1-\nobreak T)$.
\end{proof}

In \S\ref{ssect-cTp} we shall prove:
\begin{prop}\label{prop-KH1-sa}
The maps $\bold r$ and $\bold q$ give the following isomorphisms.
\[ (\K_s^0)^\uparrow\stackrel{\bold r}\lra H^1\bigl(\gen T;\V\om
s,\V\sa s\bigr)\,,\quad
(\K_s^0)^\uparrow\stackrel{\bold q}\lra H^1\bigl(\gen T;\W\om s,\W\sa
s\bigr)\,. \]
\end{prop}


\subsubsection{Semi-analytic vectors with simple
poles}\label{ssect-smp}
We turn to a $G$-module between $\V\om s$ and $\V \fs s$ that obtained
by replacing the condition of smoothness at the singularities by the
condition ``simple'', which allows simple singularities:
\begin{defn}\label{V(1)def}\indind{$\V\smp s$; $\W\smp s$}{V(1)def}We
define $\V\smp s$ as the space of $f\in \V\fs s$ such that for each
$\x\in \bsing (f)$ the function $x\mapsto c_\x(x) f^\prj(x)$ is
smooth at~$\x$, where $c_\x$ is a local coordinate on $\proj\RR$
at~$\x$, e.g., $c_\infty(x) = \frac1x$, and $c_\x(x) = x-\x$ if
$\x\in \RR$. We define $\W\smp s$ as the space of those elements of
$\W\fs s$ for which $\rho_s f\in\V\smp s$.
\end{defn}\smallskip


The spaces $\V\smp s$ and $\W\smp s$\indlbl{rWsmp}{$\tpoiss s$;
$\rho_s$} are isomorphic $G$-submodules of $\V\fs s$ respectively
$\W\fs s$, by $\tpoiss s$ and~$\rho_s$.


We have the following generalization of Proposition~\ref{prop-Tinv0}:
\begin{prop}\label{prop-Tinv1}The spaces $(\V\smp s)^T$ and
$(\W\smp s)^T$ are zero for $s\neq\frac12$.
\end{prop}
\begin{proof}Consider the expansion
$\ph(t) \sim |t|^{-2s} \sum_{n=-1}^\infty c_n t^{-n}$ at $\infty$ in
the line model, and insert the $T$-invariance. If $c_l$ is the first
non-zero coefficient, it follows that $(l+\nobreak 2s)c_l=0$.
\end{proof}

\begin{prop}\label{prop-kerav-(1)}Let $s\neq\frac12$. For
$\ph \in \V\om s$ the following statements are equivalent:
\begin{enumerate}
\item[a)] $\av T(\ph)=0$.
\item[b)] $\av T^+(\ph) = \av T^-(\ph)$.
\item[c)] There exists $f\in \V\smp s$ such that
$f\,|\,(1-\nobreak T) = \ph$.
\end{enumerate}
If these statements holds, then $f$ in c) is unique, and is equal to
$\av T^\pm(\ph)$.
\end{prop}
\begin{proof}The equivalence of a) and~b) is clear. If a) and b) hold,
then apply the asymptotic behavior in \eqref{asympt} to conclude that
$f= \av T^+(\ph)=\av T^-(\ph) \in \V\smp s$. Conversely if
$f \in \V\smp s$ satisfies $f\,|\,(1-\nobreak T)=\ph\in \V\om s$,
then $f-\av T^+(\ph)$ has for $x\uparrow\infty$ an asymptotic
behavior as indicated in \eqref{asympt}. Since
$f-\av T^+(\ph) \in (\V\smp s)^T =\{0\}$
(Proposition~\ref{prop-Tinv1}), this implies $f=\av T^+(\ph)$. Proceed
similarly for $\av T^-(\ph)$.
\end{proof}




In \S\ref{ssect-cTp} we shall prove:
\begin{prop}\label{prop-KH1-smp}If $s\neq \frac12$, the maps $\bold r$
and $\bold q$ give isomorphisms
\[ \K_s^\uparrow\stackrel{\bold r}\lra H^1\bigl(\gen T;\V\om s,\V\smp
s\bigr)\,,\quad \K_s^\uparrow\stackrel{\bold q}\lra H^1\bigl(\gen
T;\W\om s,\W\smp s\bigr)\,. \]
\end{prop}



\subsubsection{Semi-analytic vectors with support condition on the
singularities}\label{ssect-wdg} For a cohomological characterization
analogous to Propositions \ref{prop-KH1-sa} and~\ref{prop-KH1-smp} of
the much larger spaces $\bold r ( \E_s^\uparrow)^T$ and
$\bold q ( \E_s^\uparrow)^T$ we need to introduce yet another space
of semi-analytic vectors.


To show the need for this larger space, we consider $\ph$ representing
an element of $\V\om s$, in the line model. The corresponding
  function $\ph^\prj$ in the projective model is holomorphic on a
neighborhood of $\proj\RR$ in $\proj\CC$. Hence $\ph$ is holomorphic
at least on a strip $|\im\z|<\e$ and on half-planes $\re \z>\frac1\e$
and $\re \z<-\frac1\e$ for some small positive~$\e$. For
$s\neq\frac12$, the one-sided averages
\[ \av T^+(\ph)(\z) \= \sum_{n=0}^\infty \ph(\z+n),\qquad \av
T^-(\ph)(\z) \= - \sum_{n=-\infty}^{-1}\ph(\z+n)
\]
are both holomorphic on the strip $|\im\z|<\e$. Furthermore,
$\av T^+(\ph)$ is also holomorphic on $\re\z>\frac1\e$, and
$\av T^-(\ph)$ on~$\re\z<-\frac1\e$, provided $\e$ is sufficiently
small. Suppose now that $\av T(\ph)$ extends holomorphically as an
element of $\hol(\CC)^T$.
(Proposition~\ref{prop-bt-par} shows that this is the case if
$\ph=r_T$ associated to $u\in \E_s^T$.) Then
$\av T^+ (\ph)= \av T(\ph) + \av T^-(\ph)$ has a holomorphic
extension to the half-plane $\re\z<-\frac1\e$. Hence
$\av T^+ (\ph) \in \V\fs s$, and also $\av T^- (\ph) \in \V\fs s$
have representatives with large domains containing both a left and a
right half-plane. They are elements of the space $\V\wdg s$ that we
now start to define.

\begin{defn}\label{def-wn}Let $F\subset \proj\RR$ be finite. We call a
set $\Om \subset \proj\CC$ an \indind{excised
neighborhood}{def-wn}\emph{excised neighborhood} of
$\proj\RR\setminus F$ if it contains a set of the form
\be U \,\setminus\, \bigcup_{\x\in F} W_\x\,,
\ee
where $U$ is a (usual) neighborhood of $\proj\RR$ in $\proj\CC$ and
where $W_\x$ is the set containing $\x$ and the sectors in $\uhp$ and
$\lhp$ between two geodesic half-lines with final point~$\x$.
\end{defn}
\begin{figure}[ht]
\[\setlength\unitlength{.8cm}
\begin{picture}(10,8)(0,-4)
\put(0,0){\line(1,0){10}}
\put(3.1,.1){$\x_1$}
\put(5.1,.1){$\x_2$}
\put(4,.1){$\Om$}
\put(.5,-1){$\Om$}
\put(9.3,.1){$\Om$}
\put(3,0){\circle*{.1}}
\put(5,0){\circle*{.1}}
\put(3,1){\vector(0,-1){.5}}
\put(2.8,1.2){$W_{\x_1}$}
\put(3,-1){\vector(0,1){.5}}
\put(2.8,-1.6){$W_{\x_1}$}
\put(4,3){$W_\infty$}
\put(4.2,3.5){\vector(0,1){.6}}
\put(4,-3.3){$W_\infty$}
\put(4.2,-3.5){\vector(0,-1){.6}}
\thicklines
\put(2.5,0){\oval(1,1)[r]}
\put(3.5,0){\oval(1,1)[l]}
\put(4.5,0){\oval(1,1)[r]}
\put(5.5,0){\oval(1,1)[l]}
\put(2,-1){\oval(1,1)[tl]}
\put(3.5,.5){\line(1,0){1}}
\put(3.5,-.5){\line(1,0){1}}
\put(2,-.5){\line(1,0){.5}}
\put(1.5,-1){\line(0,-1){3}}
\put(1,1){\oval(1,1)[lb]}
\put(1,.5){\line(1,0){1.5}}
\put(.5,1){\line(0,1){3}}
\put(8.5,1){\oval(1,1)[rb]}
\put(9,1){\line(0,1){3}}
\put(5.5,.5){\line(1,0){3}}
\put(8.5,-1){\oval(1,1)[rt]}
\put(9,-1){\line(0,-1){3}}
\put(5.5,-.5){\line(1,0){3}}
\end{picture}
\]
\caption{An excised neighborhood~$\Om$ of $\proj\RR
\setminus\{\x_1,\x_2,\infty\}$.}\label{fig-sau1}
\end{figure}
In the upper or lower half-plane, the sets $W_\infty$ in this
definition are the region between two vertical half-lines. For
$\x\in \RR$ sets $W_\x$ are the regions between to half-circles
through~$\x$ with centers on $\RR$ to the left and the right of~$\x$.
See Figures \ref{fig-sau1} and~\ref{fig-3d} for sketches of excised
neighborhoods.
\begin{figure}[ht]
\begin{center}\[\setlength\unitlength{2cm}
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\qbezier(-0.71,0.71)(-1.,0.41)(-1.,0.)
\qbezier(-1.,0.)(-1.,-0.41)(-0.71,-0.71)
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\qbezier(0.,-1.)(0.41,-1.)(0.71,-0.71)
\qbezier(0.71,-0.71)(1.,-0.41)(1.,0.)
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\end{picture}
\]
\end{center}
\caption{An excised neighborhood of
$\uc\sm\{e^{-\pi i/4},1,e^{\pi ni/4},i\}$ for the disk
model.}\label{fig-3d}
\end{figure}


\begin{defn}\label{wdgdef}\indind{$\V\wdg s$}{wdgdef}
 Let $F\subset \proj\RR$ be finite. We say that $\ph \in \V\fs s[F]$
 satisfies the condition ``$\wdgwdg$'' if $\ph$ is represented by an
 element of $\hol(\Om)$ for an excised neighborhood of
 $\proj\RR\sm F$.
\end{defn}
This means that $\V\wdg s$ is the direct limit $ \limind\Om \hol(\Om)$
where $\Om$ runs over excised neighborhoods of $\proj\RR\sm F$ with
$F$ finite. Figure~\ref{figVs} depicts the relation of $\V\wdg s$
with other principal series subspaces that we have defined up till
now. The space $\V\smp s$ is defined only for $s\neq\frac12$. The
definition of $\V\wdg s$ makes sense for $s=\frac12$ as well.
\begin{figure}[ht]
\[\xymatrix@R=.3cm@C=.4cm{\V\om s \ar@{^{(}->}@<-.4ex>[r]
\ar@{^{(}->}@<-.4ex>[rdd]
& \V\sa s \ar@{^{(}->}@<-.4ex>[r] \ar@{^{(}->}[rd]
& \V\infty s \ar@{^{(}->}@<-.4ex>[r]
& \V{-\infty}s \ar@{^{(}->}@<-.4ex>[r]
& \V{-\om}s\\
&& \V\smp s \ar@{^{(}->}@<-.4ex>[rd]
\\
& \V\wdg s \ar@{^{(}->}@<-.4ex>[rr]
&&\V\fs s } \]
\caption{Subspaces of the principal series space $\V{}s$.
}\label{figVs}
\end{figure}

The space $\V\wdg s$ is a $G$-invariant subspace of $\V\fs s$. The
elements $\av T^\pm(\ph)$ discussed in the introduction of this
subsection are in $\V{\om,\,\wdgwdg} s [\infty]$. We have even more:

\begin{prop}\label{prop-phavhol}Let $s\neq \frac12$. For
$\ph \in \V\om s$ the following statements are equivalent:
\begin{enumerate}
\item[a)] $\av T(\ph) \in \hol(\CC)^T$.
\item[b)] $\av T^+(\ph)$ and $\av T^-(\ph)$ are elements of
$\V{\om,\wdgwdg} s[\infty]$.
\end{enumerate}
\end{prop}
\begin{proof}We have already discussed the implication
a)$\;\Rightarrow\;$b).

For a)$\;\Leftarrow\;$b) we suppose that $\av T^+(\ph)$ and
$\av T^-(\ph)$ are in $\V{\om,\wdgwdg} s[\infty]$. By construction,
they are in $\V\om s[\infty]$. They are given by holomorphic
functions on a region
$\{\z\;:\; |\im\z|<\e\} \cup \{\z\;:\; |\re \z|>\e^{-1}\}$ for some
$\e>0$. On $|\im\z|<\e$ we have $\av T(\ph) = \av T^+
(\ph)- \av T^-(\ph)$. So $\av T(\ph)$ extends holomorphically to
$\re\z>\e^{-1}$, and the invariance under $T$ stays valid by analytic
continuation. The $T$-invariance on a right half-plane implies that
$\av T(\ph)$ extends holomorphically to~$\CC$.
\end{proof}
\medskip

We will now define the condition ``$\wdgwdg$'' for the behavior of
sections of $\W\fs\s$ at singularities:
\begin{defn}\label{Wwdg}\indlbl{Wwdg}{$\W\wdg s$}An element of
$\W\fs s$ satisfies the condition ``$\wdgwdg$'' at the points of the
finite set $F\subset\proj\RR$ if it has a representative of the form
$f(z) = \left( \frac y{|z+i|^2}\right)^s a^\prj(z)$ on $\Om\cap\uhp$,
where
\begin{enumerate}
\item[a)] $\Om$ is an excised neighborhood of $\proj\RR\sm F$ for some
finite set~$F$.
\item[b)] $a^\prj$ is real analytic on~$\Om$.
\end{enumerate}
\end{defn}

The transverse Poisson transformation gives a $G$-equivariant
isomorphism
\indlbl{tp-wdg}{$\tpoiss s$}\[\tpoiss s:\V\wdg s \lra \W\wdg s\,.\]
To see this we use the integral representations of $\tpoiss s $ and
its inverse in Theorem~4.7 in~\cite{BLZ}. To apply this we note that
the intersection $\Om \cap\bar \Om$ of an excised neighborhood $\Om$
of $\proj\RR\sm F$ is again an excised neighborhood
of~$\proj\RR\sm F$, for finite sets $F\subset\proj\RR$.


Propositions \ref{prop-KH1-sa} and~\ref{prop-KH1-smp} can be viewed
partly as specializations of the following result:
\begin{thm}\label{thm-H1Wow-char}If $s\neq\frac12$, then the maps
$\bold r$ and $\bold q$ give isomorphisms
\[ (\E_s^\uparrow )^T \stackrel{\bold r}\lra H^1\bigl(\gen T;\V\om
s,\V\wdg s\bigr)\,,\quad
(\E_s^\uparrow )^T\stackrel{\bold q}\lra H^1\bigl(\gen T;\W\om
s,\W\wdg s\bigr)\,. \]
\end{thm}
We shall give a proof in~\S\ref{ssect-cTp}.

\begin{defn}\label{Gwdg-def}We define $\G^\wdgwdg_s $ as the space of
$f\in C^2(\uhp)$ that are in $\E_s(\uhp\cap \Om)$ for an excised
neighborhood~$\Om$ of $\proj\RR$ minus a finite set, and we
put\indind{$\G^\wdgwdg_s\,,\G^\wdg_s$}{Gwdg-def}
\[ \G^\wdg_s \= \bigl\{ f\in \G^\wdgwdg_s\;:\; f\text{ represents an
element of }\W\fs s \bigr\}\,.\]
\end{defn}
The minimal closed set $X\subset\uhp$ such that $f\in \E_s(\uhp\sm X)$
is the set of singularities $\sing(f)$.

\emph{Examples:} The function $i_{s,0}(z)=y^s$ is an element of $\E_s$
that represents an element of $\W\om s(\RR)$, also called $i_{s,0}$,
and $\bsing (i_{s,0})=\{\infty\}$. So $i_{s,0}\not\in \G^\om_s$. We
have $i_{s,0}(z) = \left( \frac y{|z+i|^2}\right)^s
\allowbreak \left( x^2+(y+1)^2\right)^s$. We conclude that
$a^\prj(z) =  \left( x^2+(y+1)^2\right)^s$ is real analytic on $\CC$.
Hence $i_{s,0}\in \G^\wdg_s$, and
$\sing\bigl(i_{s,0}\bigr)=\emptyset$,
$\bsing\bigl(i_{s,0}\bigr) = \{ \infty\}$. To get more examples of
elements of $\W\wdg s$, we consider $\ph\in \V\om s$ such that
$\av T(\ph)\in \hol(\CC)^T$, as in Proposition~\ref{prop-phavhol}. If
$s\neq\frac12$, then $\tpoiss s \av T^+(\ph)=\av T^+ (\tpoiss s \ph)$
and $\tpoiss s \av T^-(\ph)=\av T^-(\tpoiss s \ph)$ are elements of
$\W\wdg s$.

For general $h\in \W\om s$, the average $\av T h$ may very well have
singularities in horizontal strips in the upper half-plane. Then the
average is not in $\G^\wdg_s$.

The example of $i_{s,0}$ shows that if $f\in \G^\wdg_s$ represents
$h\in \W\wdg s$, then the set $\bsing(h)$ can be larger than
$\overline{\sing(f)}\cap\p\uhp$. On the other hand, the zero element
$n\in \W\om s$ satisfies $\bsing(n) = \emptyset$. It may be
represented as an element of $\W\wdg s$ by any $f\in C^2(\uhp)$ that
is equal to $0$ outside the region $|x|\leq 1$, $y\geq1$, and equal
to $1$ on $|x|\leq \frac12$, $y\geq 2$. In this example,
$\overline{\sing(f)} \cap\p\uhp$ is larger than $\bsing(h)$.
Definitions \ref{Wwdg} and~\ref{Gwdg-def} imply that we can always
represent $h\in \W\wdg s$ by an element $f\in \G^\wdg_s$ such that
$\overline{\sing(f)}\cap\p\uhp \subset \bsing(h)$.
\medskip

The exact sequence \eqref{NGWex} extends to an exact diagram
\be\label{PW-diagr}
\begin{array}{ccccccccc}
&&0&&0&&0\\
&&\downarrow&&\downarrow&&\downarrow\\
0 &\lra& \N^\om &\lra& \G^\om_s &\lra& \W\om s &\lra& 0\\
&&\downarrow&&\downarrow&&\downarrow\\
0 &\lra& \N^\wdg &\lra& \G^\wdg_s &\lra& \W\wdg s &\lra& 0
\end{array}
\ee
We recall that $\N^\om = C_c^2(\uhp)$, and define
\indlbl{Nwdg}{$\N^\wdg$}$\N^\wdg$ as the kernel in the lower row. The
support of an element of $\N^\wdg$ need not be compact; it may
contain regions between geodesic half-lines to the same point
of~$\p\H$. Siegel domains of $\Gmod=\PSL_2(\ZZ)$ are examples of such
sets.



\begin{lem}\label{lem-WT0}The spaces $(\W\wdg s)^T\cong(\G^\wdg_s)^T $
are equal to~$ \I s$, and the space $(\N^\wdg)^T$ is zero.
\end{lem}
\begin{proof}If $f\in (\G^\wdg_s)^T$, then the set $\bsing (f)$ is a
$T$-invariant finite subset of~$\proj\RR$, and hence is contained in
$\{\infty\}$. The set $\sing (f)$ is also $T$-invariant. It is
contained in the union of a compact set and finitely many vertical
regions. Hence $\sing (f)=\emptyset$ and $f\in \E_s^T$. Since $f$
represents an element of $\W\om s[\infty]$, it is in $\I s$. If
$f\in (\N^\wdg)^T$, then $\sing (f)=\emptyset$ implies $f=0$.

We are left with the proof of $(\W\wdg s)^T \cong (\G^\wdg_s)^T$.
Clearly, each element of $(\G^\wdg_s)^T=\I s$ represents an element
of $(\W\wdg s)^T$. Restriction of a given $h\in (\W\wdg s)^T$ gives
$\rho_s h \in ( \V \om  s[\infty])^T$. Hence $h$ has a representative
$f\in \E_s(\uhp\sm W_\e)$, where
\be\label{Weps-def} W_\e \=\{z\in \uhp\;:|\re z|\leq \e^{-1},\, \im
z\geq \e\}\ee
for some $\e>0$. By extending $f$ as a $C^2$-function on $W_\e$, we
obtain a representative $f\in \G^\wdg_s$. So
$f\,|\,(1-T) \in \N^\wdg$. After diminishing $\e$, we have
$f(z)=f(z+1)$ on $\uhp\sm W_\e$. Since $f$ represents an element of
$\W\om s[\infty]$, it has a Fourier expansion with only
$i_{s,2\pi n}$, and hence is in $\I s^\downarrow$. This expansion
converges on~$\uhp$ and defines another representative of $h$, which
is in~$\I s$.
\end{proof}



\begin{lem}\label{lem-Pow-Po}Suppose that $\hat q \in \G^\wdg_s$
satisfies:
\begin{enumerate}
\item[a)] $\sing (\hat q)$ is a compact subset of~$\H$.
\item[b)] $\hat q\,|\,(1-T) \in \G^\om_s$.
\end{enumerate}
Then there exists $p\in \I s$ such that $\hat q-p \in \G^\om_s$.
\end{lem}
\begin{proof}From a) it follows that $\hat q$ represents an element
$q$ in the space $\F_s$ defined in \eqref{F-def}. The direct sum
decomposition \eqref{F-decomp} implies that there are unique
$p\in \E_s$ and $f\in \W\om s$ such that $q=p+f$. Condition b)
implies that $q\,|\,(1-T)\in \W\om s$. Hence
$p\,|\,(1-T) = q\,|\,(1-T)-f\,|\,(1-T) \in \W\om s\cap\E_s=\{0\}$. So
$p\in \E_s^T$, and $p=q-f \in \W\wdg s$. As in the proof of the
previous lemma, this implies $p\in \I s$. Now,
$\hat q-p \in \G^\wdg_s$ is a lift of~$f\in \W\om s$, for which
$\sing(\hat q-p) = \sing(\hat q)$ is compact. Hence
$\hat q-p\in \G^\om_s$.
\end{proof}


\subsubsection{$T$-invariant eigenfunctions and
cohomology}\label{ssect-cTp}
In this subsection we prove Theorem~\ref{thm-H1Wow-char} and
Propositions \ref{prop-KH1-sa} and~\ref{prop-KH1-smp}. We treat these
three proofs in parallel, since the statements are closely
related.\smallskip




\sssect{Images}First we consider $u\in (\E_s^\uparrow )^T$ and
$s\neq\frac12$. We represent $\bold r u$ by the cocycle $r$
determined by $r_T(\z) = \int_{z_0-1}^{z_0}[u,R(\z;\dv)^s]$ with a
suitable base point~$z_0$. Relation~\eqref{rel-avT}, Lemma
\ref{lem-uupbt} and Proposition~\ref{prop-phavhol} show that
$r_T = \av T^+(r_T)\,|\,(1-\nobreak T) $ is in the space
$\V\wdg s\,|\,(1-\nobreak T)$. So $\bold r$ maps $(\E_s^\uparrow)^T$
to the subspace $H^1(\gen T;\V\om s,V^\wdg_s)$. Applying the
transverse Poisson transform, we see that
$\bold q (\E_s^\uparrow)^T $ is contained in the space
$ H^1(\gen T;\W\om s,\W\wdg s)$.

If $u\in \K_s^\uparrow$, then Lemma~\ref{lem-uupbt} shows that
$\av T(r_T)=0$. Hence $\av T^+(r_T)=\av T^-(r_t)$, and then
$\av T^+(r_T) \in \V\smp s$ by Proposition~\ref{prop-kerav-(1)}. This
show that
\[\bold r \K_s^T \subset H^1(\gen T;\V\om s,\V{\fs\!,\,
\wdgwdg\,\&\,\smpsmp} s)\subset H^1(\gen T;\V\om s,\V\wdg s)\,.\]
Again we apply $\tpoiss s$ to get the analogous statement
for~$\bold q$. (The space $\V{\fs\!,\, \wdgwdg\,\&\,\smpsmp} s$ is
equal to $\V\wdg s \cap \V\smp s$, since at the singularities both
the conditions ``$\wdgwdg$'' and ``$\smpsmp$'' are imposed.)

We have seen in~\eqref{iminf} that
$\bold q (\K_s^0)^T \subset H^1(\gen T;\W\om s,\W\sa s)$, even if
$s=\frac12$. Since the restriction map $\rho_s$ is an isomorphism on
$\W\om s$ and $\W\sa s$, the corresponding statement for $\bold r$
follows. The integrals in \eqref{Vominfint} and \eqref{Wominfint}
show that $\bold r$ and $\bold q$ map $(\K_s^0)^\uparrow$ to
$H^1(\gen T;\V\om s,\V{\fs\!,\,\wdgwdg\,\&\,\infty}s )$, respectively
$H^1(\gen T;\W\om s,\W{\fs\!,\,\wdgwdg\,\&\,\infty} s)$.



\sssect{Comparison result}For Propositions \ref{prop-KH1-sa}
and~\ref{prop-KH1-smp} it is important to have:
\begin{lem}\label{lem-comp-w}
\beqabc
\begin{align}\label{ia}
H^1(\gen T;\V\om s,\V{\fs\!,\,\wdgwdg\,\&\,\infty}s ) &\= H^1(\gen
T;\V\om s,\V\sa s)\,,\\
\label{ib}
 H^1(\gen T;\W\om s,\W{\fs\!,\,\wdgwdg\,\&\,\infty}s ) &\= H^1(\gen
 T;\W\om s,\W\sa s)\,, \end{align}
and if $s\neq\frac12$
\begin{align}
\label{ic} H^1(\gen T;\V\om s,\V{\fs\!,\,\wdgwdg\,\&\,\smpsmp} s ) &\=
H^1(\gen T;\V\om s,\V\smp s)\,,\\
\label{id}
 H^1(\gen T;\W\om s,\W{\fs\!,\,\wdgwdg\,\&\,\smpsmp} s) &\= H^1(\gen
 T;\W\om s,\W\smp s)\,
\end{align}
\eeqabc
\end{lem}
\begin{proof}We have $H^1(\gen T;\V\om s,\V\smp s) 
\supset H^1(\gen T;\V\om s,\V{\fs\!,\,\wdgwdg\,\&\,\smpsmp} s)$.
Conversely, if $\ps_T \in \V\om s$ is of the form
$\ps_T=f\,|\,(1-\nobreak T)$ with $f\in \V\smp s$, then
$\av T(\ps_T)=0$
(Proposition~\ref{prop-kerav-(1)}), and hence
$\av T^+(\ps_T),\av T^-(\ps_T)\in \V\wdg s$
(Proposition~\ref{prop-phavhol}). This gives~\eqref{ic}. For
\eqref{ia}, we proceed similarly, with use of
Proposition~\ref{prop-kerav-inf} to obtain $\av T(\ps_T)=0$.


The transverse Poisson transformation provides us with an injection
\[ \tpoiss s: \V{\fs\!,\,\wdgwdg\,\&\,\smpsmp} s \longrightarrow
\W{\fs\!,\,\wdgwdg\,\&\,\smpsmp} s\,,\]
and a bijection
\[ \tpoiss s:\V\smp s \longrightarrow \W\smp s\,.\]
The resulting commuting diagram
\[ \xymatrix{H^1(\gen T;\V\om s,\V{\fs\!,\,\wdgwdg\,\&\,\smpsmp} s )
\ar[r]^(.53){=\;\eqref{ic}} \ar[d]^{\tpoiss s}
& H^1(\gen T;\V\om s,\V\smp s)
\ar[d]^{\tpoiss s}_\cong
\\
H^1(\gen T;\W\om s,\W{\fs\!,\,\wdgwdg\,\&\,\smpsmp}s)
\ar@{^{(}->}@<-.4ex>[r]
& H^1(\gen T;\W\om s,\W\smp s)
} \]
shows the equality in \eqref{id}. For \eqref{ib} we proceed similarly.
\end{proof}


\sssect{Injectivity}This is Proposition~\ref{prop-rinj}.

\sssect{Surjectivity in Theorem \ref{thm-H1Wow-char}} Since $\V\wdg s$
and $\W\wdg s$ are isomorphic it suffices to prove that for
$s\neq\frac12$
\[ \bold q : \bigl( \E_s^\uparrow\bigr)^T
\lra H^1\bigl( \gen T;\W\om s,\W\wdg s \bigr)\]
is surjective.

We recall that $\bold q$ associates to $u\in (\E_s^\uparrow)^T$ the
cohomology class $q_T + \W\om s\,|\,(1-\nobreak T)$, where $q_T$ can
be represented by $\tilde q_T \in \G^\om_s$ given by
$\tilde q_T(z)= \int_{z_0-1}^{z_0}[u,q_s(\dv,z)]$ outside a small
neighborhood of the line segment from $z_0-1$ to~$z_0$. The class
does not depend on the choice of $z_0$ with $\im z_0$ sufficiently
large. In this proof we will mainly work with representatives in
$\G^\om_s$ and $\N^\om$. See the diagram~\eqref{PW-diagr}.
\medskip

For the proof of the surjectivity we start with $f\in \S\wdg s$ such
that $h := f\,|\,(1-\nobreak T)$ belongs to $\W\om s$. The aim is to
construct $u\in (\E_s^\uparrow)^T$ whose associated function $q_T$
satisfies $h-\frac1{\pi i} q_T \in \W\om s\,|\,(1-\nobreak T)$.

Let $\hat f\in \G^\wdg_s$ and $\hat h\in \G^\om_s$ be representatives
of $f$ and $h$, respectively. Since the set $\bsing (f)$ is finite
and $T$-invariant, it is contained in $\{\infty\}$. We choose $N>2$
sufficiently large and $\e\in (0,1)$ sufficiently small to achieve
the following situation:
\be
\setlength\unitlength{.5cm}
\begin{array}{ccc}
\sing\bigl(\hat h\bigr) \;\subset \; [-N,N]\times i[\e,N]
&\quad& \sing \bigl(\hat f\bigr)
\;\subset \; [-N,N]\times i[\e,\infty)
\\
\begin{picture}(6,5.8)(-3,-.6)
\put(-4,0){\line(1,0){8}}
\put(-3.25,-.7){$-N$}
\put(2.9,-.7){$N$}
\put(-.18,3.15){$iN$}
\put(-.18,.75){$i\e$}
\put(-3,0){\circle*{.2}}
\put(3,0){\circle*{.2}}
\put(0,.6){\circle*{.2}}
\put(0,3){\circle*{.2}}
\put(-2,1.6){(singularities)}
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\put(-3,.6){\line(0,1){2.4}}
\put(3,.6){\line(0,1){2.4}}
\end{picture}
&&
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\put(2.9,-.7){$N$}
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\end{picture}
\end{array}
\ee

The difference $k=\hat h-\hat f\,|\,(1-T)$ is an element of $\N^\wdg$.
It satisfies
\be\label{kavTh} \supp k \subset [-N-1,N]\times i[\e,\infty)\,.\ee
We set
\be u(z) \= \av T(k)(z)\qquad\text{ for }\im z>N\,.
\ee
This will turn out to represent the element of $(\E_s^\uparrow)^T$
corresponding to the cocycle on $\gen T$ given by~$h$. We prove this
in several steps.

\begin{lem}\label{lem-AkE}The function $u$ represents an element of
$\bigl(\E_s^\uparrow\bigr)^T$.
\end{lem}
\begin{proof}The average $\av T k$ is given by a locally finite sum.
For $z=x+i y\in [-1,1]\times i(0,\infty)$ we have
\[ \av T(k)(z) \= \sum_{n=-N-2}^{N+1} k(z+n)
\= \hat f(z+N+2)-\hat f(z-N-2)+ \sum_{n=-N-2}^{N+1} \hat h(z+n)\,. \]
The terms $\hat f(z+N+2)$ and $\hat f(z-N-2)$ have no singularities in
the region $\im z=y>\e$, $|x|\leq 1$, and $\hat h$ has no
singularities in the region $y>N$. Hence $\av T(k)$ is a
$\ld_s$-eigenfunction of~$\Dt$ on the region $y>N$, $|x|<1$. Since it
is $1$-periodic, it is a $\ld_s$-eigenfunction on $y>N$.
\end{proof}


We take $y_0>N+2\e$, and define
\be\label{hqTd}
\hat q_T (z) \= \int_{iy_0-1}^{iy_0} [u,q_s(\dv,z)] \ee
\twocolwithpictr{outside the box $[-1-\e,\e]\times i[y_0-\e,y_0+\e]$,
and extend $\hat q_T$ inside the box as a $C^2$-function. Thus we
obtain $\hat q_T \in \G^\om_s$ representing $q_T$. Our aim is to show
that $\hat h- \frac1{\pi i} \hat q_T \in \G^\om_s\,|\,(1-\nobreak T) 
+ \N^\om$. }{\setlength\unitlength{.7cm}
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\put(-1,1){\circle*{.08}}
\put(-1.2,.6){$\scriptstyle -1$}
\put(-.05,.6){$\scriptstyle 0$}
\put(0,1){\circle*{.08}}
\put(-1.5,2.5){$\hat q_T $ given by}
\put(-1.1,1.9){integral}
\put(-1,4){\circle*{.08}}
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\end{picture}
}\medskip

We apply Proposition~\ref{prop-udecomp-up}, which writes
\be u\= \frac1{\pi i}\,\bigl( u^\uparrow-u^\downarrow\bigr)
\=\frac1{\pi i }\,\Bigl( \av T^\uparrow(\hat q_T)
- \av T ^\downarrow (\hat q_T)
\Bigr)\,,\ee
with $u^\uparrow:=\av T^\uparrow(h) \in \K_s^\uparrow$ equal to
$\av T (\hat q_T)$ on the region $\im z>y_0+\e$ and
$u^\downarrow:=\av T^\downarrow(\hat q_T)\in \I s$ equal to
$\av T(\hat q_T)$ on $\im z<y_0-\e$. Both functions do not depend on
the choice of $y_0$.
\smallskip

The next step is to relate $\av T(\hat q_T)$ to $\av T(k)$,
$\av T(\hat h)$ and $\hat f$. To do this, we use the functions $p_+$
and $p_-$ in the next lemma:
\begin{lem}\label{lem-qpm}The following two functions $p_\pm$ belong
to $\I s$:
\[ p_+ \= \av T^+(\hat h) - \av T^+(k) - \hat f\,,\qquad p_- \= \av
T^-(\hat h) - \av T^-(k) - \hat f\,.\]
\end{lem}
\begin{proof}
$\av T^+(k)$ is given by a locally finite sum, and $\av T^+(\hat h)$
can be understood in regularized sense, since $\hat h \in \G^\om_s$.
Hence $p_+$ is defined on $\uhp$, except for its singularities, which
occur on a locally finite union of curves in the region
$\left\{ z\in \uhp\;:\; x\leq N\,,\; y\geq \e\right\}$. {}From
$(\hat h - \nobreak k)\,|\,\av T^+|(1-\nobreak T) =
\hat h-k = \hat f\,|\,(1-\nobreak T)$ it follows that
$p_+|(1-\nobreak T) =0$. Since $p_+$ has no singularities in the
region $x>N$, the $T$-invariance implies that there are no
singularities at all. Hence $p_+ \in \E_s^T$.

On $0<y<\e$, we have $k=0$. Moreover, $\av T^+(\hat h)$ represents an
element of $\W\om s[\infty]$, and
$\hat f\in \G^{\om,\,\wdgwdg}_s[\infty]$. So $p_+$ represents an
element of $\W\om s(\RR)$. Hence it is in~$\I s$.

The case of $p_-$ goes similarly.
\end{proof}

\begin{lem}\label{lem-qhl}
\[ \frac1{\pi i} \av T(\hat q_T) \=
\begin{cases}
\av T(\hat h)&\text{ on}\quad\im z>y_0+\e\,,\\
p_+-p_- &\text{ on}\quad\im z<y_0-\e\,.
\end{cases}\]
\end{lem}
\begin{proof}
On the region $\im z>N$ we have
\[ u \= \av T \= \av T(\hat h)+p_--p_+\,. \]
Lemma~\ref{lem-avWas} implies that
$\av T(\hat h) = c_1 y^{1-s} + \oh(y^{-s})$ as $y\rightarrow\infty$
for $|x|\leq 1$. By $T$-invariance this estimates holds for all $x$,
and shows that the restriction of $\av T(\hat h)$ to $y>N$ is in
$\K_s^\uparrow$. Lemma~\ref{lem-qpm} gives $p_+-p_-\in \I s$. Hence
we have obtained the terms in the decomposition of $u$ in
$(\E_s^\uparrow)^T = \K s^\uparrow + \I s$ given in
Proposition~\ref{prop-udecomp-up}:
\[ u^+ \= \av T^\uparrow(\hat q_T) \= \pi i\, \av T(\hat h)\,,\quad
u^- \= \av T^\downarrow(\hat q_T) \= \pi i\,
(p_+-p_-)\,.\]
The averages $\av T^\uparrow (\hat q_T)$ and
$\av T^\downarrow(\hat q_T)$ are both given by $\av T(\hat q_T)$ on
the regions $\im z>y_0+\e$ and $\im z< y_0-\e$, respectively.
\end{proof}

We will choose a function $g\in C^2(\uhp)$ that will turn out to
satisfy
$g\,|\,(1-\nobreak T) \in \hat h -\frac1{\pi i} \hat q_T +\N^\om$.
First we prescribe $g$ on the union of the following three
overlapping regions:
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\]
\be g(\tau)\=\begin{cases}
\av T^+(\hat h) - \frac1{\pi i} \av T^+(\hat q_T) - p_+ &\text{ on
}H\,,\\
\hat f + \av T^+(k-\frac1{\pi i}\hat q_T)
&\text{ on }R\,,\\
\hat f + \av T^-(k-\frac1{\pi i}\hat q_T) + p_--p_+ &\text{ on }L\,.
\end{cases}
\ee

To see that this is possible, we have to do some checking on the
intersections. On $y<\e$, i.e., on $R\cap L$, we check:
\begin{alignat*}2 \av T^+(k-\frac1{\pi i}\hat q_T) &-\bigl( \av T^-(k-
\frac1{\pi i} \hat q_T) + p_--p_+\bigr)
\\
&\= 0 - \frac1{\pi i}\av T(\hat q_T)
+p_+-p_-&&\text{ since }k=0\text{ on }y<\e
\\
&\= 0 -p_++p_-+p_--p_+ \=0&\quad&\text{ by Lemma \ref{lem-qhl}}\,.
\end{alignat*}

On $H\cap R$, i.e., for $y>y_0+\e$ and $x>N$:
\[ \hat f + \av T^+(k-\frac1{\pi i} \hat q_T)
- \bigl( \av T^+(\hat h-\frac1{\pi i} \hat q_T) - p_+\bigr)\=0
\quad\text{ by Lemma \ref{lem-qpm}}\,. \]

On $H\cap L$, i.e., for $y>y_0+\e$ and $x<-N$:
\begin{alignat*}2 \hat f
&+ \av T^-\bigl(k-\frac1{\pi i}\hat q_T\bigr)
+p_--p_+
- \Bigl( \av T^+\bigl(\hat h-\frac1{\pi i}\hat q_T\bigr) - p_+\Bigr)\\
&\= \av T^-(\hat h) - \frac1{\pi i} \av T^-(\hat q_T) - \av T^+(\hat
h) + \frac1{\pi i} \av T^+(\hat q_T)
&\quad&\text{ by Lemma \ref{lem-qpm}}\\
&\= - \av T(\hat h) + \frac1{\pi i} \av T(\hat q_T) \=0 &&\text{ by
Lemma \ref{lem-qhl}}\,.
\end{alignat*}
Thus, we have $g$ on $H\cup R \cup L$, i.e, only on the exterior of
the box $[-N,N]\times i[\e,y_0+\nobreak\e]$. We extend it by $0$ on
the box, and change it on an $(\e/2)$-neighborhood of the boundary to
bring it into~$C^2$.


\begin{lem}\label{lem-g}We have $g\in \G^\wdg_s$ and
$g\,|\,(1-T) \in \G^\om_s$.
\end{lem}
\par\noindent\emph{Proof. }The function
$\hat f + \av T^+(k-\nobreak \frac1{\pi i}\hat q_T) =
\hat f - \frac1{\pi i} \av T^+ (\hat q_T)$ on $R \cap L$ represents
an element of $\W\om s(\RR)$. The singularities of $g$ are contained
in the box $[-N,N] \times i[\e,y_0+\e]$. Hence $g\in \G^\wdg_s$.

On the region~$R$:
\[ g\,|\,(1-T) \= \hat f\,|\,(1-T)
+k-\frac1{\pi i} \hat q_T \= \hat h
- \frac1{\pi i} \hat q_T\,. \]
Since $\hat h-\frac1{\pi i} \hat q_T\in \G^\om_s$, the equality
$g\,|\,(1-\nobreak T) = \hat h-\frac1{\pi i} \hat q_T$ extends to the
region in $\uhp$ outside the box
$[-N-\nobreak 1,N] \times i[\e,y_0+\nobreak \e]$. So there is
$g_1 \in \N^\om$ such that
\be\label{g1T}
g\,|\,(1-\nobreak T) = \hat h-\frac1{\pi i} \hat q_T + g_1\in
\G^\om_s\,.\qed\ee
%\end{proof}

For the given $h\in \W\om s$ and $f\in \W\wdg s$ with
$h=f\,|\,(1-\nobreak T)$ we have given $u\in (\E_s^\uparrow)^T$, and
have given in \eqref{hqTd} a representative $\hat q_T$ of the
corresponding $q_T\in \W\om s$ which determines the cocycle
$\bold q u$. In Lemma~\ref{lem-g} we see that
$\hat h - \frac1{\pi i} \hat q_T 
\in \N^\om + \G^\wdg_s\,|\,(1-\nobreak T)$. Hence
$h-\frac1{\pi i} q_T \in \W\wdg s\,|\,(1-\nobreak T)$. This completes
the proof of the surjectivity in Theorem~\ref{thm-H1Wow-char}.

\sssect{Surjectivity in Proposition~\ref{prop-KH1-smp}}We need only
prove the surjectivity of~$\bold q$. Applying
Theorem~\ref{thm-H1Wow-char} to a given $c\in H^1(\gen
T;\W\om s,\W\wdg s\cap \W\smp s)$, we obtain a unique
$u\in (\E_s^\uparrow)^T$ such that $\bold q u =c$. We have to check
that $u\in \K_s^\uparrow$.

In the proof of the surjectivity that we have just given, we now have
the additional information that $f\in \W\wdg s \cap \W\smp s$. Hence
$\av T(h)=0$ (by Proposition~\ref{prop-kerav-(1)}), and
$\av T(\hat h)$ vanishes near~$\RR$. Since $k$ also vanishes
near~$\RR$, the difference $p_+-p_-\in \I s$ in Lemma~\ref{lem-qpm}
vanishes near~$\RR$ as well, and hence $q_+-p_-=0$ everywhere. In
Lemma~\ref{lem-qhl} we see that $\av T(\hat q_T)^\downarrow=0$. Hence
\eqref{kavTh} takes the form
$u = \frac1{\pi i} \av T(\hat q_T)^\uparrow\in \nobreak \K_s^\uparrow$.


\sssect{Surjectivity in Proposition~\ref{prop-KH1-sa}} Now $s$ may be
equal to~$\frac12$. This forces a further review of the proof of the
surjectivity of $\bold q$ for Theorem~\ref{thm-H1Wow-char}.

We start with $f\in\W\wdg s \cap \W\sa s$ with
$h=f\,|\,(1-\nobreak T) \in \W\om s$. This implies that
$h(z) = \bigl( \frac y{|z+i|^2}\bigr)^s \,  a^\prj(z)$ with
$a^\prj(\infty)=0$. So $\av T^\pm(h)$ and $\av T^\pm(\hat h)$
converge without the need for regularization. The absolute
convergence of $\av T^\pm(k)$ is clear anyhow. We have by
Lemma~\ref{lem-AkE} a function $u\in (\E_s^\uparrow)^T$ given on
$y>N$ by $\av T(k)$. We cannot yet use the one-sided averages of
$\hat q_T$, but still have $p_+$ and $p_-$ in $\E_s^T$ in
Lemma~\ref{lem-qpm}. Proposition~\ref{prop-kerav-inf} and an
application of the isomorphisms $\rho_s:\W\sa s\lra\V\sa s$ and
$\tpoiss s : \V\sa s\lra \W\sa s$ imply that
$\av T^+(\hat h) = \av T^-(\hat h)$ near~$\RR$. Near $\RR$, we also
have $k=0$. So $p_+-p_- = \av T(\hat h) - \av T(k) $ vanishes first
near~$\RR$, and then everywhere on~$\uhp$. On $y>N$ we have
$\av T(\hat h) = \av T(k) + p_+-p_- = \av T(k)$ as in the proof of
Lemma~\ref{lem-qhl}. Hence $\av T(\hat h) = u$ on $y >N$.
Lemma~\ref{lem-avWas} gives $u(z) = \av T(\hat h)(z) \ll y^{- s}$ as
$y\rightarrow\infty$. So $u$ can have only terms with $k_{s,2\pi n}$,
$n\neq0$, in its Fourier expansion. Thus, $u\in(\K_s^0)^\uparrow$.


With $\hat q_T$ as in \eqref{hqTd}, we use \eqref{Wominfint} and
Proposition~\ref{proppsCi} to write
$\hat q_T \in h_{iy_0}\,|\,(T-\nobreak 1) + \N^\wdg$, with
$h_{iy_0}\in \G^\sa_s$, $\sing (h_{iy_0})\subset i[y_0,\infty)$ and
$\bsing(h_{iy_0})\subset\{\infty\}$. In particular, $(\rho_s\hat
q_T)^\prj(\infty)=0$, and $\av T^+(\hat q_T)$ and $\av T^-(\hat q_T)$
converge absolutely. Now we can proceed as before.



\skipbetweenchapters
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%   Chapter 4   groups with cusps                   %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\sect{Maass forms and semi-analytic cohomology: groups with
cusps}\label{ch-IV}


In this chapter we start the generalization of the results for the
modular group $\PSL_2(\ZZ)$ mentioned in the introduction to general
cofinite discrete subgroups $\Gm\subset G$ with cusps. We will prove
those statements in Theorems~\ref{thmB} and~\ref{thmC} that concern
cohomology groups with semi-analytic coefficients. The results
concerning smooth ($C^\infty$) and differentiable
($C^p$ for some $p\in\NN$) coefficients will be proved in
Chapter~\ref{ch-V}.


In Section~\ref{sect-MFsac} we consider the isomorphisms
$\Mf_s^0(\Gm) \cong \Hpar^1(\Gm;\V\om s,\V\sa s)$
(in Theorem~\ref{thmB}),
$\E_s^\Gm \cong \Hpar^1(\Gm;\V\om s,\V\wdg s)$
(in Theorem~\ref{thmC}), and a similar isomorphism using the space
$\V\smp s\supset \V\sa s$ introduced in \S\ref{ssect-smp}. The method
of proof is the same as that followed for cocompact groups in
Chapter~\ref{ch-II}. The presence of cusps makes it necessary to look
at geometrical models for cohomology again, especially in connection
with parabolic cohomology. That is the subject of
Section~\ref{sect-cohparcoh}, where we also discuss an interpretation
of our approach to parabolic cohomology in terms of sheaf cohomology.

In Section~\ref{sect-pcmpc} we prove the isomorphisms
$\Hpar^1(\Gm;\V\om s,\V\sa s)  \cong \Hpar^1(\Gm;\V\sa s)$
(in Theorem~\ref{thmB}) and $\Hpar^1(\Gm;\V\om s,\V\wdg s)\cong
\Hpar^1(\Gm;\V\wdgcp s ) $ (in Theorem~\ref{thmC}), where
$\V\wdgcp s$ consists of the elements of $\V\wdg s$ that have
singularities in cusps only. This requires an analysis of the set of
singularities of cocycles. In \S\ref{recapC} we give a recapitulation
of the proof of Theorem~\ref{thmC}.


This chapter generalizes results of~\cite{LZ}, where Maass cusp forms
on $\SL_2(\ZZ)$ were related to ``period functions''. The link with
the period function is discussed in Section~\ref{sect-Mf}, where we
also give some general definitions, and in Section~\ref{sect-pf}. In
\eqref{btdef} we gave a holomorphic function associated to a
$\ld_s$-eigenfunctions invariant under the parabolic element
$T=\matc1101$ of~$G$. Such linear maps to the space of $1$-periodic
holomorphic functions on~$\CC$ can also be defined for
 $\Gm$-invariant eigenfunctions. That is the subject of
Section~\ref{sect-Mfhol}.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Maass forms}\label{sect-Mf}
Throughout this chapter the group
\indlbl{gam1}{$\Gm$}$\Gm \subset G=\PSL_2(\RR)$ is assumed to have
cusps. For such groups we discuss several spaces of Maass forms and
general invariant eigenfunctions, which coincide for cocompact groups
discussed in Chapter~\ref{ch-II}.

The image of the map from invariant eigenfunctions to cohomology with
values in the space $\V\om s$ of analytic vectors in the principal
series is contained in a mixed parabolic subgroup, of which we will
give a preliminary definition in this section.
\medskip

Here the upper half-plane $\uhp$ is the natural model of~$\H$. A
discrete subgroup $\Gm\subset G$ is called
\indlbl{cof}{cofinite}\emph{cofinite} if the quotient
$\Gm\backslash \uhp$ has finite volume for the measure induced by the
invariant measure $d\mu$ on~$\uhp$. The
\indlbl{cusp}{cusp}\emph{cusps} of~$\Gm$ are points $\kp\in \proj\RR$
for which the isotropy subgroup
\indlbl{itrsg}{$\Gm_\kp}$\Gm_\kp=\left\{\gm\in \Gm\;:\; \gm\kp=\kp\right\}$
is non-trivial, and hence infinite cyclic with a parabolic generator.
We denote by
\indlbl{cu}{$\cu$}$\cu$ the set of cusps of~$\Gm$. This set depends
on~$\Gm$. It is infinite, but consists of finitely many $\Gm$-orbits.

For each $\kp\in \cu$ we fix
\indlbl{gkap}{$g_\kp$}$g_\kp\in G$ such that $g_\kp \infty=\kp$ and such
that
\indlbl{pikap}{$\pi_\kp$}$\pi_\kp = g_\kp T g_\kp^{-1}$ generates
$\Gm_\kp$, with $T= \matc1101$. This leaves some freedom in the choice
of the $g_\kp$. We arrange the $g_\kp$ such that
$g_{\gm \kp} \in \gm g_\kp T^\ZZ$ within each $\Gm$-orbit of cusps.

The standard example is the
\indlbl{mg}{modular group; $\Gmod$}modular group $\Gmod=\PSL_2(\ZZ)$,
generated by $T$ and
\indlbl{S}{$S = \matr0{-1}10$}$S=\matr0{-1}10$, with relations
$S^2=(TS)^3=1$. Its set of cusps $\cu=\proj\QQ$ forms one
$\Gmod$-orbit.


\subsubsection{Notations and terminology}\label{ssect-tM} We call the
elements of $\E_s^\Gm$ \indlbl{ie}{invariant
eigenfunction}\emph{invariant eigenfunctions}, and reserve the
notation
\indlbl{Mf}{$\Mf_s(\Gm)$}$\Mf_s(\Gm)$ for the finite-dimensional space
$(\E_s^{-\infty})^\Gm$ of invariant eigenfunctions with polynomial
growth, whose elements we call \indlbl{Maf}{Maass form}\emph{Maass
forms}. An invariant eigenfunction $u\in \E_s^\Gm$ has
\indlbl{pogr}{polynomial growth}\emph{polynomial growth} if and only
if
\beqabc
\be u\left( g_\kp(x+iy) \right) \ll y^a \text{ as } y\rightarrow\infty
\text{ for some } a \in\RR,\text{ uniform in }x,\text{ for all
}\kp\in\cu\,. \ee
Inside $\Mf_s(\Gm)$ the space
\indlbl{cf}{Maass cusp form; cusp form;$\Mf_s^0(\Gm)$}$\Mf^0_s(\Gm)$
of \emph{(Maass) cusp forms} is determined by the stronger condition
of \indlbl{qd}{quick decay}\emph{quick decay} at all cusps:
\be \label{qui-dec}
u\left( g_\kp(x+iy) \right) \ll y^a \text{ as } y\rightarrow\infty
\text{ for all } a \in \RR,\text{ uniform in }x,\text{ for all
}\kp\in\cu\,. \ee
\eeqabc
By the $\Gm$-invariance, it suffices that these growth conditions hold
for one representative $\kp$ of each $\Gm$-orbit of cusps. In
\cite{LZ} we used $\Mf_s$ to denote the space which we now call
$\Mf_s^0(\Gmod)$. For cocompact groups, the spaces $\Mf_s^0(\Gm)
\subset\Mf_s(\Gm) \subset\E_s^\Gm$ coincide.

Let $u\in \E_s^\Gm$. For each $\kp\in \cu$, the function
$u\,|\,g_\kp:z\mapsto u(g_\kp z)$ is in $\E^T_s$, and has a Fourier
expansion \eqref{Fourexp-par}, with coefficients $A_n(u\,|\,g_\kp)$
and $B_n(u\,|\,g_\kp)$, and also $a_0(u\,|\,g_\kp)$ and
$b_0(u\,|\,g_\kp)$. The space $\Mf_s(\Gm)$ is characterized by
$B_n(u\,|\,g_\kp) =0$ for $n\neq0$ for all~$\kp$, and the space
$\Mf^0_s(\Gm)$ by the additional requirement
$A_0(u\,|\,g_\kp)=B_0(u\,|\,g_\kp)=0$ (for $s\neq\frac12$), or
$a_0(u\,|\,g_\kp)=b_0(u\,|\,g_\kp)=0$
(for all $s$), for all~$\kp$. The form of the Fourier expansion implies
that we can weaken \eqref{qui-dec} by replacing ``for all
$a\in \RR$'' by ``for some $a < \min(\re s,1-\re s)$''.


For $s\neq\frac12$ we define the space
\indlbl{Mf1}{$\Mf_s^1(\Gm)$}$\Mf_s^1(\Gm)\subset \E_s^\Gm$ by the
condition that $B_n(u\,|\,g_\kp)=0$ for all $\kp\in \cu$ and all
$n\in \ZZ$. We have
$\Mf_s^0(\Gm) \subset \Mf_s^1(\Gm) \subset \Mf_s(\Gm)$. It may happen
that $\Mf_{1-s}^1(\Gm) \neq \Mf_s^1(\Gm)$ ( e.g., if $\Gm=\Gmod$ and
$2s$ is a zero of the Riemann zeta function), whereas $\Mf_s^0(\Gm)$,
$\Mf_s(\Gm)$ and $\E_s^\Gm$ are invariant under $s\mapsto 1-s$.


\subsubsection{Invariant eigenfunctions and parabolic
cohomology}\label{ssectMaf-pc}We start with an example. A $1$-cocycle
$\ps$ on the modular group $\Gmod$ with values in a right
$\QQ[\Gmod]$-module~$V$ is, in the group model of cohomology,
determined by $\ps_T$ and $\ps_S$ on the generators
\indlbl{Sgn1}{$S$ in mod. group}\indlbl{Tgn1}{$T$ in mod.
group}$S=\matr0{-1}10$ and $T=\matc1101$, subject to the relations
$\ps_S\,|\,(1+\nobreak S) = 0$ and $\ps_{TS}\,|\,(1+TS + TSTS)=0$
(and $\ps_{TS}=\ps_T\,|\,S+\ps_S$). There are various possibilities of
normalization. We may for instance require that $\ps_S=0$, which can
be arranged by subtracting $da$ from $\ps$, with $a\in V$ given by
$\frac12 \, \ps_S$. Another normalization is by arranging
$\ps_{TS}=0$; then $\ps$ is determined by $\ps_S=\ps_T$ satisfying
$\ps_S\,|\,S=-\ps_S$. For the cocycle $r$ in~\eqref{rdef} associated
to an invariant eigenfunction, the former normalization is arranged
by choosing the base point $z_0$ equal to~$i$, and the latter by
choosing $z_0=\frac12(1+i\sqrt 3)$.

In general, it is impossible to choose $\ps$ in its cohomology class
such that $\ps_T=0$. However, in the case of a cusp form
$u\in \Mf_s^0(\Gmod)$, and $r$ as in \eqref{rdef}, we know from
\eqref{rhh} that $r_T $ is of the form $f_{z_0}\,|\,(T-\nobreak1)$,
with $f_{z_0}\in \V\sa s $. Thus, subtracting $df_{z_0}$ from~$r$, we
obtain a $\V\sa s$-valued cocycle satisfying $\ps_T=0$. This cocycle
is determined by its value $\ps_S$. Since
$(\V\sa s)^T \subset (\V\infty s)^T=\{0\}$
(Proposition~\ref{prop-Tinv0}), this cocycle is unique. This motivates
the following definition:
\begin{defn}\label{def-pcG}Let $V\subset W$ be $\QQ[\Gm]$-modules. We
define the \emph{mixed parabolic cohomology group} $\Hpar^1(\Gm;V,W)$
as $\Zpar^1(\Gm;V,W) / B^1(\Gm;V)$, where\indlbl{defpcG}{mixed
parabolic cohomology group; $\Hpar^i(\Gm;V,W)$; $\Zpar^i(\Gm;V,W)$;
mixed parabolic cocycle}
\[ \Zpar^1( \Gm;V,W ) \=\bigl\{ \ps \in Z^1\left( \Gm;V\right)\;:\;
\ps_\pi \in W\,|\,(\pi-1)\text{ for all parabolic }\pi \in \Gm
\bigr\} \]
is the space of \emph{mixed parabolic cocycles}.


We define the \indlbl{pc}{parabolic cohomology group;parabolic
cocycle; $\Hpar^i(\Gm;V)$}\emph{parabolic cohomology group}
$\Hpar^1(\Gm;V)$ as $\Hpar^1(\Gm;V,V)$, and call the elements of
$\Zpar^1(\Gm;V,V)$ \emph{parabolic cocycles}.
\end{defn}

It suffices to impose the parabolic condition only for $\pi=\pi_\kp$
with $\kp$ running through a set of representatives of
$\Gm\backslash\cu$. The mixed parabolic cohomology group
$\Hpar^1(\Gm;V,W)$ is the kernel of the natural map
\be H^1(\Gm;V) \lra \bigoplus_{\kp\in \Gm\backslash\cu}
H^1(\Gm_\kp;W)\,.\ee
We may view the group $H^1(\gen T;V,W)$ in~\eqref{mcd} as a mixed
parabolic cohomology group.

Returning to the case $\Gm=\Gmod$, we see that the $\V\sa s$-valued
cocycle $\ps$ with $\ps_T=0$ associated above to a Maass cusp form
$u$ satisfies $\ps_S \in \V{\wdg\,\&\,\infty} s$, since
$f_{z_0}\in \V{\wdg\,\&\,\infty}s $.
(See Definition~\ref{wdgdef} for~$\V\wdg s$.)
Actually, the singularities of
$\ps_S= r_S - f_{z_0}\,|\,(S-\nobreak 1)$ can occur only in $\infty$
and~$0=S\infty$. So the cocycle $\ps$ has values in
$\V{\fsn\!,\, \wdgwdg\,\&\,\infty} s$, where we use the following
definition:
\begin{defn}\indlbl{Vom0}{$\V{\om^0}s$,
$\V{\fsn\!,\,\mathrm{cond}} s$, $\V\smpcp s$, $\V\wdgcp s$}Let
$\V{\om^0}s$ be the $\Gm$-submodule consisting of the $f\in \V\fs s$
such that $\bsing (f) \subset\cu$. With a condition ``cond'' imposed
on the singularities, we put
$\V{\fsn\!,\,\mathrm{cond}}s=\V\fsn s\cap \V{\fs,\!,\,\mathrm{cond}}s$.

For $\W\ast s$ we follow the same convention.
\end{defn}
We recall that elements of $\V\fs s$ can have a finite number of
arbitrary singularities on $\p\uhp=\proj\RR$, those of $\V\smp s$
(Definition~\ref{V(1)def})
have a ``simple pole''
( i.e., $\tau\mapsto (\tau-\tau_0)(\text{smooth})$ at real points
$\tau_0$), and the singularities of elements of $\V\wdg s$ occur
outside an excised neighborhood of $\proj\RR$ minus a finite set
(Definition~\ref{def-wn}).

\begin{prop}\label{prop-im-coh}
The injective maps $\bold r$ and $\bold q$ determined by
\eqref{pqrdef} have images in the following equal cohomology groups.
\beqabc
\begin{alignat}2 \bold r \,\Mf_s^0(\Gm)
&\;\subset\; \Hpar^1(\Gm;\V\om s,\V\sacp s)
&\=&\Hpar^1(\Gm,\V\om s,\V\sa s)\,,\\
\bold q \, \Mf_s^0(\Gm) &\;\subset\; \Hpar^1(\Gm;\W\om s,\W\sacp s)
&\=&\Hpar^1(\Gm,\W\om s,\W\sa s)\,.
\end{alignat}
If $s\neq\frac12$, then we also have
\begin{alignat}2 \bold r \, \Mf_s^1(\Gm)
&\;\subset\; \Hpar^1(\Gm;\V\om s,\V\smpcp s)
&\=&\Hpar^1(\Gm,\V\om s,\V\smp s)\,,\\
\bold q \, \Mf_s^1(\Gm) &\;\subset\; \Hpar^1(\Gm;\W\om s,\W\smpcp s)
&\=&\Hpar^1(\Gm,\W\om s,\W\smp s)\,,\\
\bold r \,\E_s^\Gm &\;\subset\; \Hpar^1(\Gm;\V\om s,\V\wdgcp s)
&\=&\Hpar^1(\Gm,\V\om s,\V\wdg s)\,,\\
\bold q \, \E_s^\Gm &\;\subset\; \Hpar^1(\Gm;\W\om s,\W\wdgcp s)
&\=&\Hpar^1(\Gm,\W\om s,\W\wdg s)\,.
\end{alignat}
\eeqabc
\end{prop}
\begin{proof}By definition, $\Hpar^1(\Gm;\V\om s,\V\wdgcp s) \subset
\Hpar^1(\Gm\;\V\om s,\V\wdg s)$. Consider a parabolic cocycle
$c\in \Zpar^1(\Gm;\V\om s,\V\wdg s)$ and a parabolic element
$\pi=\pi_\kp$, $\kp\in \cu$. Then
$c_\pi \,|\,(1-\nobreak\pi) \in \V\om s$ implies that the
$\pi$-invariant set $\bsing(c_\pi)$ is contained in~$\{\kp\}$. So
$[c] \in \Hpar^1(\Gm;\V\om s,\V\wdgcp s)$. This proves the equality
of these cohomology groups. The same proof works for the other
equalities of parabolic cohomology groups.

For the inclusion $\bold r\, \E_s^\Gm$ in
$\Hpar^1(\Gm;\V\om s,\V\wdg s)$ we take $u\in \E_s^\Gm$, and suppose
that $s\neq \frac12$. Let $z_0\in \uhp$ be the base point in the
definition in \eqref{rdef} of the cocycle $r:\Gm \lra \V\om s$. Let
 $\kp\in \cu$. The value $r_{\pi_\kp}$ on the generator
 $\pi_\kp=g_\kp T g_\kp^{-1}$ of $\Gm_\kp$ satisfies
\[ r_{\pi_\kp}(\z) \= \int_{g_\kp T^{-1} g_\kp^{-1}z_0}^{z_0} [u,
R(\z;\dv)^s] \= \int_{T^{-1}g_\kp^{-1}z_0}^{g_\kp^{-1}z_0} \bigl[
u\,|\,g_\kp, R(\dv,\dv)|_{2s} \,g_\kp^{-1}\,(\z)\bigr]\,, \]
where we have used the $G$-equivariance of the Green's form
$[\dv,\dv]$ in \eqref{[]invar} and of $R(\dv;\dv)^s$
in~\eqref{Requiv}. Hence
\[ r_{\pi_\kp}\,|\,g_\kp\,(\z) \=
\int_{T^{-1}g_\kp^{-1}z_0}^{g_\kp^{-1}z_0} [ u\,|\,g_\kp,
R(\z;\dv)^s]\,,\]
which shows that $r_{\pi_\kp}\,|\,g_\kp= r^\kp_T$, where $r^\kp$ is a
cocycle on $\gen T$ that represents
$\bold r (u\,|\,g_\kp)  \in H^1(\gen T;\V\om s)$, with the base point
$g_\kp^{-1}z_0$. Theorem~\ref{thm-H1Wow-char} shows that
\[\bold r( u\,|\,g_\kp)
\in H^1(\gen T;\V\om s,\V\wdg s)\,,\]
as defined in \eqref{mcd}. Hence there is $a_\kp \in \V\wdg s$ such
that $r^\kp_T = a_\kp\,|\,(T-\nobreak 1)$, and
\[ r_{\pi_\kp} \= a_\kp \,|\,(T-1)\,|\,g_\kp^{-1} \=
(a_\kp\,|\,g_\kp^{-1})
\,|\,(\pi_\kp-1)\,. \]
This works for all $\kp\in \cu$, and hence $\bold r u \in
\Hpar^1(\Gm;\V\om s,\V\wdg s)$.

The other assertions go similarly, using also Propositions
\ref{prop-KH1-sa} and~\ref{prop-KH1-smp}, and taking into account
that $\Mf^1_s(\Gm)$ is characterized in $\E_s^\Gm$ by
$u\,|\,g_\kp \in \K_s$ for all cusps $\kp\in \cu$, and $\Mf^0_s(\Gm)$
by $u\,|\,g_\kp \in \K_s^0$ for all $\kp\in \cu$.
\end{proof}


We can use Proposition~\ref{prop-q-decomp} to reconstruct
$u \in \E_s^\Gm$ from the value $q_\pi$ of the cocycle~$q$ in
\eqref{qdef} on any parabolic $\pi\in \Gm$.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Cohomology and parabolic cohomology for groups with
cusps}\label{sect-cohparcoh}We now turn to a more geometrical
description of the cohomology of cofinite discrete groups with cusps,
like we did in \S\ref{sect-coh-cc} for cocompact discrete subgroups
of~$G$. For the standard cohomology groups we do not work with
resolutions built on $\uhp$, but on a contractible subset
$\uhp_\hy\subset\H$, where $\hy$ is a parameter. For the parabolic
cohomology, we work on $\uhp^\ast\supset\uhp$, where all cusps
of~$\Gm$ are added to~$\uhp$. Using tesselations of $\uhp^\ast$ we
will define $\Hpar^i(\Gm;V,W)$ for all $i\geq 0$, extending
Definition~\ref{def-pcG}. In Proposition \ref{prop-sh} we will relate
these mixed parabolic cohomology groups $\Hpar^i(\Gm;V,W)$ to sheaf
cohomology groups.



\subsubsection{Resolutions}\label{ssect-res-c}For $\kp\in \cu$ and
$a>0$, we define the
\indlbl{horocycle}{horocycle; $H_a(\kp)$; horocyclic disk;
$D_a(\kp)$}\emph{horocycle} $H_a(\kp)$ and the \emph{open horocyclic
disk} $D_a(\kp)$ as follows:
\twocolwithpictr{
\be \label{horocycle}
\begin{aligned}
 H_a(\kp) &\;=\;g_\kp (ia+\RR)\,,\\\
 D_a(\kp) &\;=\; \left\{ g_\kp z\;:\; \im z>a\right\}\, .
\end{aligned}\ee
$D_a(\infty)$ is a euclidean half-plane, whereas $D_a(\kp)$ is a
euclidean disk touching $\RR$ in $\kp$ if $\kp\in \RR$.
\indlbl{uhpsa}{$\uhp_a$}We denote
$\uhp_a = \uhp \sm \bigcup_{\kp\in \cu}D_a(\kp)$.

\pridtc We provide the \indlbl{uhpa}{extended upper half-plane;
$\uhp^\ast$}\emph{extended upper half-plane}
$\uhp^\ast = \uhp \cup\cu$
(depending on $\Gm$ via $\cu$) with its usual topology that induces
the standard topology on~$\uhp$, and has the extended horocyclic
disks $D_a(\kp)\cup\{\kp\}$, $a>0$, as a basis of open
neighborhoods\break}{
\setlength\unitlength{1cm}
\begin{picture}(5,5)(0,-.4)
\put(0,0){\line(1,0){5}}
\put(5.05,-.15){$\RR$}
\put(4,3.9){$H_a(\infty)$}
\put(2,4.8){$D_a(\infty)$}
\put(2.1,.9){$D_a(\kp)$}
\put(3.37,.3){$H_a(\kp)$}
\put(2.45,-.25){$\kp$}
\thicklines
\put(0,3.8){\line(1,0){5}}
\put(2.5,1){\circt}
\end{picture}
}\vskip-1.8ex\noindent
of $\kp\in \cu$. This topology is finer than that induced by the
inclusion $\uhp^\ast \subset \bar\uhp\subset \proj\CC$.



\sssect{Fundamental domain}We shall work with a
\indlbl{fdc}{fundamental domain; $\fd$}fundamental domain $\fd$ for
$\Gm\backslash\uhp$ that satisfies the following conditions: We
require that $\fd$ is a \indlbl{D}{Dirichlet fundamental
domain}Dirichlet fundamental domain, constructed from a base
point~$P_0$ that is not an elliptic fixed point. There is a finite
set
\indlbl{Ac}{$A$}$A=A^{-1}\subset \Gm$ such that $\al\fd$, $\al\in A$,
runs through the $\Gm$-translates of~$\fd$ that have an edge in
common with~$\fd$. We add the cusps in the closure of~$\fd$ in
$\uhp^\ast$ to~$\fd$. We require that this extended fundamental
domain meets each $\Gm$-orbit of cusps exactly once. This is possible
(\cite{Leh}, Chap.~IV, \S7G on p.~151). In particular,
\indlbl{fdcu}{$\fdcu$}$\fdcu= \fd\cap\cu$ can and often will be used
as a set of representatives for $\Gm\backslash\cu$.

The \indlbl{stfd}{standard fundamental domain}standard fundamental
domain $\fd= \{ z\in \uhp\;:\; |x|\leq \frac12\,,\; |z|\geq 1\}$ for
the modular group satisfies these requirements, with
$\fdcu = \{\infty\}$.


\sssect{Tesselations}\indlbl{t2}{tesselation} For a fundamental
domain~$\fd$ as above, we choose \indlbl{hy}{$\hy$}$\hy>0$ large
enough that all $D_\hy(\kp)$ are pairwise disjoint and that the
following geodesic segments are contained in the interior of
$\uhp_\hy$: the segments $e(P_0,\al^{-1}P_0)$ for $\al\in A$, all
segments connecting $P_0$ to the vertices of $\fd\cap \uhp_\hy$, and
all segments connecting those vertices.

In the sequel we will need tesselations of the following four types.
\begin{enumerate}
\item[i)] \emph{Tesselation of type \Dcoh. }\indlbl{Dcohc}{\Dcoh}With
$X_0=\Gm P_0$,
$X_1 = \Gm\left\{ e(P_0,\al^{-1}P_0)\;:\; \al\in A\right\}$, and
$X_2$ the collection of the closures of the connected regions
enclosed by the edges in $X_1$, we get a tesselation of a
contractible region contained in~$\uhp_\hy$. This leads to a
projective resolution $\Fpt = \QQ[X_\pnt]$, which computes the group
homology and the group cohomology of~$\Gm$.



\item[ii)] \emph{Tesselation of type \Fcoh. }\indlbl{Fcohc}{\Fcoh}The
fundamental domain $\fd$ gives a tesselation $\tess$ of $\uhp$. We
add to the edges the $\Gm$-translates of the intersections of $\fd$
with $\p D_\hy(\kp)$ for $\kp\in \fdcu$. These edges are not geodesic
segments. In this way,
$\fd= \fd_\hy \cup \bigcup_{\kp\in \fdcu} V_\kp$, where
\indlbl{fdgy}{$\fd_\hy$}$\fd_\hy = \fd\cap\uhp_\hy$, and where
\indlbl{Vkap}{$V_\kp$}$V_\kp  = \{ g_\kp z \;:\;  \im z\geq
\hy\,,\; x_\kp-1 \leq \re z\leq x_\kp\} $ for some
\indlbl{xkap}{$x_\kp$}$x_\kp\in \RR$, is a triangle with infinite height
and finite hyperbolic area, with vertices $\kp$,
\indlbl{Pkap}{$P_\kp$}$P_\kp = g_\kp(x_\kp+\nobreak i\hy) \in \p \uhp_\hy$
and
$\pi_\kp^{-1}P_\kp = g_\kp(x_\kp-\nobreak 1+\nobreak i\hy)\in \p\uhp_\hy$.


We write \indlbl{efkap}{$e_\kp=e(P_\kp,\kp)$;
$f_\kp=e(P_\kp,\pi_\kp^{-1}P_\kp)$}$e_\kp = e(P_\kp,\kp)$ and
$f_\kp=e(P_\kp,\pi_\kp^{-1}P_\kp) \subset \p H_\hy(\kp)\cap\fd$. So $e_\kp$
is a geodesic half-line and $f_\kp$ is a horocyclic segment. We have
\be \p V_\kp \= e_\kp-\pi_\kp^{-1} e_\kp - f_\kp\,.\ee
There is a finite set \indlbl{Ec}{$E$, $\gm_e$}$E$ of edges $e$ of
$\fd_\hy$ and corresponding $\gm_e\in \Gm$ such that
\be \p\fd_\hy \= \sum_{\kp\in \fdcu} f_\kp + \sum_{e\in E} e -
\sum_{e\in E} \gm_e^{-1} e\,.\ee
We denote\indind{$B$}{Efd}
\be \label{Efd}
B \= E \sqcup \{ e_\kp,\, f_\kp\;:\; \kp\in \fdcu\}\,. \ee
See Figure~\ref{figmodtess} for an illustration in the modular case.
\begin{figure}[ht]
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\]
\caption{Modular group, parts of tesselations of type \Fcoh\ (left)
and \Dcoh\ (right).} \label{figmodtess}
\end{figure}


We put\indind{$X_j^\hy$; $X_j$}{XXY}
\be\label{XXY}
\renewcommand\arraystretch{1.5}
\begin{array}{cll}
\qquad\quad&X_0^\hy \= \Gm\left\{ \text{vertices of
}\fd_\hy\right\}\,,& X_0\= X_0^\hy \sqcup \cu\,,\\
&X_1^\hy\= \Gm \left\{ f_\kp\;:\; \kp\in \fdcu\right\} \cup \Gm E \,,&
X_1\= X_1^\hy \sqcup \Gm \left\{ e_\kp\;:\; \kp\in \fdcu\right\}\,,\\
&X_2^\hy \= \Gm (\fd_\hy)\,,& X_2\= X_2^\hy \sqcup \Gm\left\{
V_\kp\;:\; \kp\in \fdcu\right\}\,.
\end{array}
\ee
Here, and in the sequel, we consider the elements of the sets $X_j$ as
compact subsets of~$\uhp^\ast$. We have arranged that all elliptic
fixed points are elements of $X_0^\hy$.

The translates of $\fd_\hy$ form a tesselation $\tess$ of the
contractible space $\uhp_\hy$, and \indlbl{Ffcc}{
$\Fpt^{\tess,\hy}$}$\Fpt^{\tess,\hy}= \QQ[X_\pnt^\hy]$ is a
projective resolution of the $\QQ[\Gm]$-module~$\QQ$. It is contained
in the chain complex
\indlbl{F-Tcalc}{$\Fpt^\tess$}$\Fpt^\tess= \QQ[X_\pnt]$, which is not
projective, due to the cusps in $\cu \subset X_0$. The set $B$ in
\eqref{Efd} is a $\QQ[\Gm]$-basis of $F_1^\tess$, and the following
set is a $\QQ[\Gm]$-basis of~$F_2^\tess$:
\be \{\fd_\hy\} \cup \left\{ V_\kp\;:\; \kp\in\fdcu\right\}\,.\ee

All other tesselations that we consider
(apart from type \Dcoh), are refinements of a tesselation of
type~\Fcoh.

\item[iii)] \emph{Tesselation of type \Mcoh. }\indlbl{Mcohc}{\Mcoh}
Add to a tesselation $\tess$ of type \Fcoh\ based on a Dirichlet
fundamental domain as above the interior base point~$P_0$ and the
edges from $P_0$ to the vertices of~$\fd_\hy$. Taking
$\Gm$-translates of the new points and edges we obtain a refinement
of~$\tess$, which turns out to be a triangulation. See
Figure~\ref{figmodFD}.

\item[iv)] \emph{Tesselation of type \Mcoha. }\indlbl{Mcohac}{\Mcoha}
Take the common refinement $\tess$ of tesselations of type \Mcoh\ and
\Dcoh\ built on the same Dirichlet fundamental domain. Add all
$\Gm$-translates of the geodesic half-lines from $P_0$ to the cusps
in $\fdcu$ and the resulting additional vertices on the edges~$f_\kp$.
We call the resulting triangulation a tesselation of type~\Mcoha. See
Figure~\ref{figmodFD}.
\end{enumerate}
\begin{figure}[ht]
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\end{center}
\caption{Modular group, part of tesselations of type \Mcoh\ (left) and
\Mcoha\ (right).} \label{figmodFD}
\end{figure}

\sssect{Chain complex on $\uhp^\ast$} The chain complex on $\uhp^\ast$
gives the resolution \indlbl{hcohc}{$\Fpt^\hcoh$;
$\Fpt^{\hcoh,\hy}$}$F_i^\hcoh = \QQ\bigl[
(\uhp^\ast)^{i+1}\bigr]$, with boundary maps, augmentation and group
action as in~\eqref{chH}. It has a subcomplex
$F^{\hcoh,\hy}_i = \QQ[\uhp_\hy^{i+1}]$.




\subsubsection{Cohomology groups for~$\Gm$}\label{ssect-Tcoh}For all
tesselations $\tess$ that refine a tesselation of type~\Fcoh, the
complex $\Fpt^{\tess,\hy}$ is a subcomplex of $\Fpt^\tess$ that gives
a projective resolution of~$\QQ$ and can be used to compute the
cohomology groups $H^\hpnt(\Gm;V)$. For this purpose, we can also use
the complex corresponding to a tesselation of type~\Dcoh, and the
complex~$\Fpt^{\hcoh,\hy}$.

If there are cusps, then $H^2(\Gm;V)=\{0\}$. In model~\Fcoh,
$F_2^{\tess,\hy}$ is generated by $\fd_\hy$. For a cocycle~$c$, the
freedom in $c(\fd_\hy) $ is determined by a coboundary
$ d b(\fd_T) = \sum_{e\in E} b(e)\,|\,(1-\gm_e) 
+ \sum_{\kp\in \fdcu}b(f_\kp)$. The $b(f_\kp)\in V$ can be freely chosen.


\subsubsection{Parabolic cohomology}\label{ssect-par-coh} We will base
the definition of parabolic cohomology groups on parabolic
resolutions defined below. For the definition of mixed parabolic
cohomology groups we use resolutions based on a tesselation of type
\Fcoh\ or a refinement of such a tesselation. To put these
definitions in context, we shall prove
(Proposition~\ref{prop-sh}) that the resulting cohomology groups are
isomorphic to certain cohomology groups in sheaf cohomology.
Moreover, we will show that in dimension~$1$ these cohomology groups
are isomorphic to those in Definition~\ref{def-pcG}.




\sssect{Parabolic resolutions}For all resolutions obtained from a
refinement of a tesselation of type \Fcoh\ there is an exact sequence
\[ \lra F_3 \stackrel{\p_3}\lra F_2 \stackrel{\p_2}\lra F_1
\stackrel{\p_1}\lra F_0 \stackrel\e\lra \QQ \lra 0\]
of $\QQ[\Gm]$-modules such that
\begin{enumerate}
\item[a)] $F_0$ has a set $G_0$ of generators over $\QQ[\Gm]$, such
that for each $x\in G_0$ the subgroup $\Gm_x\subset\Gm$ fixing~$x$ is
either finite, or equal to $\Gm_x$ with $x\in \cu$.
(In the modular case, with a tesselation of type \Fcoh\ based on the
standard fundamental domain we may take $G_0 $ consisting of $i$,
$\frac{1+i\sqrt 3}2$, $P_\infty$ and~$\infty$.)
\item[b)] For each $\kp\in \cu$ the $\QQ$-subspace
$\left( F_0\right)^{\pi_\kp}$ has dimension~$1$, and the
augmentation~$\e$ is non-trivial on this subspace.
(In the modular case, $(F_0)^T = \QQ(\infty)$.)
\item[c)] The $F_i$, $i\geq 1$, are free $\QQ[\Gm]$-modules.
\end{enumerate}
In resolutions coming from a tesselation of $\uhp^\ast$, we have
$F_i=0$ for $i\geq 3$.

We call any resolution with the properties a)--c) a
\indlbl{pres}{parabolic resolution}\emph{parabolic resolution}
of~$\QQ$. For the moment, we have only the example of resolutions
based on a refinement of a tesselation of type~\Fcoh. In
Chapter~\ref{ch-VI}, we will use another example, obtained by taking
tensor products.

Most of the properties of projective resolutions carry over:
\begin{lem}\label{lem-apcc}If $f_\pnt:\Fpt\lra \Fpt$ and
$g_\pnt:\Fpt\lra \Fpt$ are augmentation preserving chain maps of a
parabolic resolution $\Fpt$ of~$\QQ$, then they are
homotopic.\end{lem}
\begin{proof} In dimension~$0$, we have for each $\kp\in \cu$ a unique
element $b_\kp\in F_0$ such that $\pi_\kp b_\kp=b_\kp$, and $\e b_\kp=1$.
This forces $f_0b_\kp = g_0b_\kp$. {}From $\e\circ f_0 = \e \circ g_0$,
we conclude that there is a $\Gm$-equivariant map $h_0:F_0\lra F_1$
such that $\p_1h_0=f_0-g_0$. It satisfies $h_0 b_\kp=0$ for all
$\kp\in \cu$. The further construction of a homotopy between $f_\pnt$
and $g_\pnt$ goes in the same way as for projective resolutions. See
\cite{Brn}, Lemma~7.4 in Chap.~I.\end{proof}
\begin{lem}\label{lem-agpcm} If $\Fpt$ and $\Fpt'$ are parabolic
resolutions of~$\QQ$, then there exists an augmentation preserving
chain map $f_\pnt:\Fpt\lra \Fpt'$.\end{lem}
\begin{proof} For each $\kp\in \cu$, we are forced to have
$f_0 b_\kp = b_\kp'$. The further construction of the $f_i$ on the
generators can be carried out as for projective resolutions. See
\cite{Brn}, \S7 of Chap.~I.
\end{proof}




\begin{defn}\label{pcdef}The
\indind{parabolic cohomology group;
$\Hpar^i(\Gm;V)$}{pcdef}\emph{parabolic cohomology groups}
$\Hpar^i(\Gm;V)$ of~$\Gm$ with values in a $\QQ[\Gm]$-module~$V$ are
the cohomology groups of the complex
\[\Hom_{\QQ[\Gm]}(\Fpt, V)\cong \map(X_\pnt,V)^\Gm\]
for any parabolic resolution $\Fpt$ of~$\QQ$.
\end{defn}
The parabolic cohomology groups for different parabolic resolutions
are canonically isomorphic.
\medskip

In the case of the parabolic resolutions $\Fpt^\tess$ based on a
refinement $\tess$ of a tesselation of type~\Fcoh, there is a
subcomplex $\Fpt^{\tess,\hy}$ which forms a projective resolution
of~$\QQ$. We use such parabolic resolutions to define the mixed
parabolic cohomology groups:
\begin{defn}\label{mpcdef}Let $V\subset W$ be an inclusion of
$\QQ[\Gm]$-modules. We define
\indind{$C^i(\Fpt;V,W)$}{mpcdef}$ C^i(\Fpt^\tess;V,W) $ to be the
space of $\Gm$-equivariant maps $c: X_i^\tess \lra W$ such that
$c\bigl( X_i^{\tess,\hy}\bigr)
\subset V$. We define coboundary maps
\indind{$d^i$}{mpcdef}$d^i : C^i(\Fpt^\tess;V,W) \allowbreak 
\lra C^{i+1}(\Fpt^\tess;V,W)$ by $d^i c(x) = (-1)^{i+1} c(\p_{i+1}x)$.
We call the cohomology groups of the resulting complex
\[ 0 \lra C^0(\Fpt^\tess;V,W)
\stackrel{d^0}\lra C^1(\Fpt^\tess;V,W)
\stackrel{d^1}\lra C^2(\Fpt^\tess;V,W) \lra 0 \]
the \indlbl{mpcg}{mixed parabolic cohomology group;
$\Hpar^i(\Gm;V,W)$; $Z^i(\Fpt;V,W)$; $B^i(\Fpt;V,W)$}\emph{mixed
parabolic cohomology groups} $\Hpar^i(\Gm;V,W)$:
\be\label{mpcgd}
\begin{aligned}
\Hpar^i(\Gm;V,W)&\= Z^i(\Fpt^\tess;V,W)/B^i(\Fpt^\tess;V,W)\,,\\
Z^i(\Fpt^\tess;V,W)&\= \ker\left( d^i: C^i(\Fpt^\tess;V,W)\lra
C^{i+1}(\Fpt^\tess;V,W)\right)\,,\\
B^i(\Fpt^\tess;V,W)&\=
\begin{cases}d^{i-1}
C^{i-1}(\Fpt^\tess;V,W)&\text{ if }i>0\,,\\
0&\text{ if }i=0\,.\end{cases}
\end{aligned}
\ee
\end{defn}

The definition is justified by Lemmas \ref{lem-apcc}
and~\ref{lem-agpcm}, which show that up to canonical isomorphisms the
space $\Hpar^i(\Gm;V,W)$ is independent of the choices made. Note
that $\Hpar^i(\Gm;V,W)=\{0\}$ for $i> 2$. In the case $V=W$ we denote
$\Hpar^i(\Gm;V,V)$ by $\Hpar^i(\Gm;V)$. Since $\Fpt^\tess$ is a
parabolic resolution, Definitions~\ref{pcdef} and~\ref{mpcdef} lead
to isomorphic parabolic cohomology groups. Finally, this definition
is a redefinition in the case $i=1$; in Proposition~\ref{prop-mpc-t}
we will see that Definitions~\ref{def-pcG} and~\ref{mpcdef} lead to
isomorphic spaces $\Hpar^1(\Gm;V,W)$.




\sssect{Dimension $0$}For all $V\subset W$, we have
$H^0(\Gm;V,W) = V^\Gm$ (use that $X_0^{\tess, \hy}\neq\emptyset$).

\sssect{Dimension $1$}Consider the newly defined $\Hpar^1(\Gm;V,W)$ in
a tesselation $\tess$ of type \Mcoh. If $c\in Z^1( F_1^\tess;V,W)$,
then $\ps_\gm =c(\gm^{-1}P_0,P_0)$ is a cocycle on $\Gm$ with values
in~$V$. For $\kp\in \cu$:
\[ \ps_{\pi_\kp} \= c(P_0,\kp)\,|\,(\pi_\kp-1)
\in W\,|\,(1-\pi_\kp)\,. \]
So $\ps$ is a parabolic cocycle. If $c=db$ is a coboundary, then
$\ps_\gm = b(P_0)\,|\,(\gm-1) $, so $\ps \in B^1(\Gm;V)$. This gives
a map from $\Hpar^1(\Gm;V,W)$ defined here to the mixed parabolic
cohomology group in Definition~\ref{def-pcG}.

Conversely, since $\Fpt^{\tess,\hy}$ is a projective resolution, there
is associated a cocycle $c\in Z^1( \Fpt^{\tess,\hy};V)$ to each
parabolic cocycle $\ps$ on~$\Gm$. For each $P \in X_0^\hy$, the
cocycle $\gm\mapsto c(\gm^{-1}P,P)$ is in the same cohomology class
as~$\ps$, and hence is parabolic. For each cusp $\kp\in \fdcu$, there
is $w_\kp\in W$ such that
$c(f_\kp)=c(\pi_\kp^{-1}P_\kp,P_\kp)=w_\kp\,|\,(\pi_\kp-1)$. Extend $c$ in
a $\QQ[\Gm]$-linear way to $X_1$ by defining $c(e_\kp)=-w_\kp$ for all
$\kp\in\fdcu$. Then $c\in Z^1(\Fpt^\tess;V,W)$ corresponds to the
parabolic cohomology class $[\ps]$. Thus, we have:
\begin{prop}\label{prop-mpc-t}The mixed parabolic cohomology group
$H^1(\Gm;V,W)$ defined in Definition~\ref{mpcdef} is isomorphic to
that defined in Definition~\ref{def-pcG}.\end{prop}


\sssect{Dimension $2$} The second mixed parabolic cohomology groups do
not necessarily vanish:
\begin{prop}If $|\Gm\backslash\cu|=1$, then for each $\kp \in \cu$:
\[ \Hpar^2(\Gm;V,W) \;\cong\; W \biggr/\biggl( W\,|\,(1-\pi_\kp) +
\sum_{\gm\in \Gm}V\,|\,(1-\gm) \biggr) \,,\]
\end{prop}
\begin{proof}We work with a tesselation of model~$\Fcoh$. Let
$\fdcu=\{\kp\}$. Any cocycle
$a\in Z^2(\Fpt^\tess;V,W) = C^2(\Fpt^\tess;V,W)$ is determined by its
values $a(\fd_\hy)\in V$ and $a(V_\kp)\in W$. The freedom consists of
adding $dc$ with $c\in C^1(\Fpt^\tess;V,W) $. Choosing $c(e)\in V$
for $e\in E$ changes $a(\fd_\hy)$ by $c(e)\,|\,(1-\gm_e)$ and leaves
$a(V_\kp)$ unchanged. These elements generate
$\sum_\gm V\,|\,(1-\gm)$. Choosing $c(e_\kp)\in W$ changes $a(V_\kp)$
by $c(e_\kp)\,|\,(1-\pi_\kp)$ and leaves $a(\fd_\hy)$ unchanged.
Finally, the choice of $c(f_\kp)\in V$ changes
$\bigl( a(\fd_T), a(V_\kp)\bigr)$ by $\bigl( c(f_\kp),-c(f_\kp)\bigr)$.
Thus, we can arrange $a(\fd_T)=0$, and get $a(V_\kp) \in W$ with
freedom in $W\,|\,(1-\pi_\kp) + \sum_{\gm\in \Gm}\,|\,V(1-\gm)$. This
completes the proof, and implies that the description is up to
isomorphism independent of the choice of $\kp$ in its
$\Gm$-orbit~$\cu$. To make this isomorphism explicit we consider
$\dt\in \Gm$, and note that $w\mapsto w|\dt$ leaves $W$ and $V$
invariant, and sends $W\,|\,(1-\pi_\kp) +
\sum_{\gm\in \Gm}V\,|\,(1-\gm) $ to
\[ W|\dt(1-\dt^{-1}\pi_\kp\dt) + \sum_\gm V |\dt(1-\dt^{-1}\gm\dt) \=
W|(1-\pi_{\dt^{-1}\kp})+\sum_\gm V|(1-\gm)
\,.\qedhere\]
\end{proof}
Along the same lines, we arrive at the following more complicated
description for the general case.   We recall, from
\S\ref{ssect-res-c}, 
that  we use a fundamental domain $\fd$ for which $\fdcu$ is a system of 
representatives of the finitely many 
$\Gm$-orbits of cusps.
\begin{prop} For any cofinite group $\Gm$
with
cusps the mixed parabolic cohomology group $\Hpar^2( \Gm;V,W)$ is
isomorphic to the space
$ \bigoplus_{\kp\in \fdcu} W $ modulo the sum of the following three
subspaces:
\[
\bigoplus_{\kp\in \fdcu} W|(1-\pi_\kp)\,,\qquad 
\bigoplus_{\kp\in \fdcu}\sum_{\gm\in \Gm} V|(1-\gm)\,,\qquad \Bigl\{
(v_\kp) \in \bigoplus_{\kp\in \fdcu} V \;:\; \sum_\kp
v_\kp=0\Bigr\}\,.
\]
\end{prop}

In the special case $V=0$, we have $\Hpar^i(\Gm;0,W)=\{0\}$ for
$i=0,1$, and
\be\Hpar^2(\Gm;0,W)
\= \bigoplus_{\kp\in\fdcu}
W\left/\left(W\,|\,(1-\pi_\kp)\right)\right.\,.\ee
If $V=W$, then
\be\label{H2par}
\Hpar^2(\Gm;V) \;\cong\; V_\Gm\,. \ee
This isomorphism is given by evaluation on the \indind{fundamental
class}{fc-c}fundamental class represented by
\be \label{fc-c}
(\fd) = (\fd_\hy) + \sum_{\kp\in \fdcu} \left(V_\kp\right)\,. \ee


Thus we have
\be
\renewcommand\arraystretch{1.4}
\begin{array}{ccccccccc}
\Hpar^0(\Gm;V,W) &=& \Hpar^0(\Gm;V)&=&H^0(\Gm;V)&=& V^\Gm \,,\\
\Hpar^1(\Gm;V) &\subset& \Hpar^1(\Gm;V,W)
&\subset& H^1(\Gm;V)\,,\\
\Hpar^2(\Gm;V,W) &\supset& \Hpar^2(\Gm;V)
&=&V_\Gm &\supset& H^2(\Gm;V)
&=&\{0\}\,.
\end{array}
\ee

\sssect{Sheaf cohomology}\indlbl{shcoh}{sheaf cohomology}We now show
that the mixed parabolic cohomology groups can be identified with
cohomology groups of certain sheaves on $\Gm\backslash\uhp^\ast$.
This will then be used to give a long exact sequence for mixed
parabolic cohomology groups
(Proposition~\ref{prop-lesmpc}).

The topological space
\indlbl{XX}{$\xg$}$\xg:= \Gm\backslash \uhp^\ast$ is compact. It
contains the open subspace
\indlbl{YY}{$\yg$}$\yg=\Gm\backslash \uhp$ and the closed subspace
$\yg_{\!\hy}:= \Gm\backslash \uhp_\hy$. We denote the projection map
by $\pi: \uhp^\ast \lra \xg$. Let $V\subset W$ be $\QQ[\Gm]$-modules.
On $\uhp^\ast$ we have the constant sheaf $W\times \uhp^\ast$ with
subsheaf $(V\times
\uhp_\hy) \cup \left(W\times(\uhp^\ast\sm \uhp_\hy) \right)$. The
group $\Gm$ acts by $(w,P)\,|\,\gm = (w\,|\,\gm,\gm^{-1}P)$. The
quotient\indind{$\sh_{V,W}$}{sheafdef}
\be\label{sheafdef} \sh_{V,W} \= \left. \biggl( (V\times \uhp_\hy)
\cup \bigl(W\times(\uhp^\ast\sm \uhp_\hy)
\bigr) \biggr) \right/\Gm\,
\ee
is a sheaf on $\xg$. For open $U\subset \xg$ the space $\sh_{V,W}(U)$
consists of all locally constant $\Gm$-equivariant functions
$\pi^{-1}(U) \lra W$ that take values in $V$ on the components of
$\pi^{-1}(U)$ that intersect~$\pi^{-1}(\yg_{\!\hy})$.

For a given $P\in \xg$, choose $z\in \uhp^\ast$ with $P=\pi z$. The
stalk $\left( \sh_{V,W}\right)_P$ is isomorphic to $V^{\Gm_z}$ if
$P\in \yg_{\!\hy}$ and to $W^{\Gm_z}$ if $P\in \xg\sm \yg_{\!\hy}$.
The isotropy group $\Gm_z$ is trivial for all but finitely many
$P\in \xg$.

\begin{prop}\label{prop-sh}Let $V\subset W$ be an inclusion of
$\QQ[\Gm]$-modules. Then
\[ \Hpar^j(\Gm;V,W) \cong H^j(\xg;\sh_{V,W})\qquad \text{for all
}j\geq 0\,.\]
\end{prop}
\begin{proof}For any refinement $\tess$ of a tesselation of
type~\Fcoh, we form for $z\in X_0$ the open set
\[\Om_z \= \{ z\} \cup\left\{ \mathring e\;:\; e\in X_1,\, z\in
e\right\} \cup \left\{ \mathring V\;:\; V\in X_2,\, z\in V
\right\}\,. \]
in~$\uhp^\ast$. By $\mathring e$\indlbl{inte}{$\mathring e$}, we mean
$e$ minus its end points, not the
(empty) interior as a subset of~$\uhp^\ast$. For $z\in X_0\cap\uhp$,
the set $\Om_z$ contains finitely many $\mathring e$ and
$\mathring V$. If $\kp\in \cu$, then $\Om_\kp$ is equal to
$D_\hy(\kp) \cup\{\kp\}$. If $\Gm_z$ is non-trivial, the set $\Om_z$
necessarily contains $\Gm_z$-equivalent points. We require that the
tesselation~$\tess$ is such that all $\Om_z$ contain no more
$\Gm$-equivalent points than necessary: If $\Gm_z=\{1\}$, then
$\Om_z$ should not contain $\Gm$-equivalent points; otherwise, if
$z_1$, $\gm ^{-1} z_1 \in \Om_z$ for $\gm\in\Gm$, then
$\gm\in \Gm_z$. We also require that $X_2$ consists of triangles. A
tesselation of type \Mcoha\ satisfies these conditions.

The set $\mathfrak A = \left\{ \pi \Om_z\;:\; z\in X_0 \right\}$ is a
finite open covering of~$\xg$. The intersection of two different
elements of $\mathfrak A$, if non-empty, contains the image
$\pi \mathring e$ for exactly one $\Gm$-orbit of $e\in X_1$. The
non-empty intersection of three different elements of $\mathfrak A$
corresponds to the $\pi \mathring V$ for exactly one $\Gm$-orbit of
elements $V\in X_2$. In this way, we can check that the complex
$C^\cdot(\Fpt,V,W)$ is isomorphic to the complex
$\left( C^i\left(\mathfrak A,\sh_{V,W}\right) \right)_i$ in \v Cech
cohomology. This implies the following isomorphism:
\[ \Hpar^i(\Gm;V,W) \,\cong\, \check H^i\left( \mathfrak A;\sh_{V,W}
\right)\,.\]
See, e.g., \cite{Har}, Chap.~III, \S4 for \v Cech cohomology.

Leray's theorem (Exercise 4.11, {\sl loc.\ cit.}) states that
$\check H^i\left(\mathfrak A;\sh_{V,W}\right) \cong
H^i\left(X;\sh_{V,W}\right)$ if
$H^k\left( U;\sh_{V,W}|_U\right)  = \{0\}$ for all intersections $U$
of elements of~$\mathfrak A$ for all $k\geq 1$. To finish the proof,
we have to check that this condition holds in the present
situation.\smallskip

We first consider a connected set $U$ that does not contain the image
of an elliptic or parabolic fixed point of~$\Gm$. It may happen that
$U$ is contained in $\yg_{\!\hy}$ or in $\xg\sm \yg_{\!\hy}$. Then
the restriction $\G= \sh_{V,W}|_U$ is the constant sheaf $\bold V$
or~$ \bold W$, and $H^i\left(U;\sh_{V,W}|_U\right)=0$ for $i\geq 1$,
since constant sheaves have trivial cohomology. The other possibility
is that $U$ is a neighborhood of $\pi\mathring e$ for some edge~$e$
contained in $\p \uhp_\hy$. Then $\G(U_1)=W$ if
$U_1\subset \xg\sm \yg_{\!\hy}$ and $\G(U_1)=V$ otherwise. The set
$U_0=U\cap \yg_{\!\hy}$ is closed in $U$. Let $k:U_0\mapsto U$ denote
the inclusion. We have an exact sequence of sheaves of $\QQ$-vector
spaces on~$U$:
\be\label{Ges} 0 \longrightarrow \G \longrightarrow \bold W
\longrightarrow k_\ast (\bold W/\bold V) \longrightarrow 0\,,\ee
where $\bold W$ is the constant sheaf given by $W$ on~$U$, and
$\bold W/\bold V$ the constant sheaf on $U_0$ given by $W/V$. For
$i\geq 1$, we have $H^i\left( U;\bold W\right)  = \{0\}$, and
$H^i \left( U;k_\ast\left( \bold W/\bold V\right) \right) 
\cong H^i\left( U_0;\bold W/\bold V \right) $ by Lemma III.2.4, {\sl
loc.\ cit.}, and hence also $H^1(U;k_\ast(\bold W/\bold V))=\{0\}$.
The long exact sequence corresponding to \eqref{Ges} starts with
\[ \begin{array}{ccccccccccccc}
0& \lra & V & \lra & W & \lra & W/V & \lra & H^1(U;\G) & \lra & H^1(
U;\bold W) &\lra\\
 & & & & & & & & & & || \\
  & & & & & & & & & & 0
\end{array}
\]
Since $W\lra W/V$ is surjective, this implies that $H^1(U;\G)=\{0\}$.
The later parts
\[ \begin{array}{ccccccc}
\lra& H^{i-1}(U;k_\ast(\bold W/\bold V) &
\lra& H^i(U;\G) &
\lra& H^i(U;\bold W) &\lra\\
& || & & & & || \\
& 0 & & & & 0
\end{array}
\]
of the long exact sequence show that $H^i(U;\G)=0$ for $i\geq 2$.
\smallskip

Suppose now that $U$ contains the image of a parabolic or elliptic
fixed point. Then $U = \pi\, \Om_z$ where $z\in \cu$ or $z$ is an
elliptic fixed point. We treat the case $z=\kp\in \cu$. The other case
goes similarly.

Let $P=\pi \, \kp$. The restriction of $\G$ to $U\sm\{P\}$ is the
constant sheaf $\bold W$. With the injection $k:\{P\} \rightarrow U$,
we have the following exact sequence of sheaves on~$U$:
\[ 0 \longrightarrow \G\longrightarrow \bold W \longrightarrow
k_\ast\left( \bold W/\bold W_\kp \right) \longrightarrow 0\,,\]
where $\bold W_\kp$ is the constant sheaf $W/W^{\Gm_\kp}$ on $\{P\}$. We
proceed as in the previous case.
\end{proof}



This proposition shows that the concept of mixed parabolic cohomology
in Definition~\ref{mpcdef} can be interpreted as sheaf cohomology.
Since a sequence of sheaves is exact if all corresponding sequences
of stalks are exact, we have:
\begin{prop}\label{prop-lesmpc}Suppose that the rows in the following
diagram of $\QQ[\Gm]$-modules are exact
\be\label{exdi}
\begin{aligned}&\xymatrix{ 0\ar[r]&V'\ar[r] \ar@<-.7ex>@{^{(}->}[d]
&V\ar[r] \ar@<-.7ex>@{^{(}->}[d] &V''\ar[r]
\ar@<-.7ex>@{^{(}->}[d]&0\\
0\ar[r]&W'\ar[r]&W\ar[r]&W''\ar[r]&0 }
\end{aligned} \ee
and that for each $\kp\in \fdcu$ the sequence
\be\label{cuex}
0\longrightarrow\left( W'\right)^{\Gm_\kp} \longrightarrow W^{\Gm_\kp}
\longrightarrow \left( W''\right)^{\Gm_\kp} \longrightarrow0
\ee
is exact as well. Then there is a long exact sequence of mixed
parabolic cohomology groups
\begin{align*} \cdots \lra
&\Hpar^{i-1}(\Gm;V'',W'') \lra \Hpar^i(\Gm;V',W') \lra
\Hpar^i(\Gm;V,W) \\
\lra
&\Hpar^i(\Gm;V'',W'')
\lra \Hpar^{i+1}(\Gm;V,W) \lra \cdots
\end{align*}
\end{prop}

Parabolic cohomology groups are sometimes defined as the kernel of the
restriction to the boundary in the Borel-Serre compactification,
e.g., in \cite{DH5}, \S2. In dimension~$1$ this leads to parabolic
cohomology groups isomorphic to the groups $\Hpar^1(\Gm;V)$ defined
here.\medskip



The category with as its objects inclusions $V\subset W$ of
$\QQ[\Gm]$-modules and the obvious morphisms is not abelian. To get a
fully satisfactory cohomological treatment of mixed parabolic
cohomology, one should extend Definition~\ref{mpcdef} to morphisms
$V \lra W$ of $\QQ[\Gm]$-modules. We refrain from carrying out this
extension, and mention only one case, which will be used in
Section~\ref{sect-pcmpc}. Let $V \hookrightarrow W$ be an inclusion
of $\QQ[\Gm]$-modules. Define $\Qcal_{W/V}$ as the quotient in the
exact sequence of sheaves on~$X$
\be \label{qshe} 0 \lra \sh_{V,W} \lra \sh_{W,W}\lra \Qcal_{W/V}\lra
0\,. \ee
So $\Qcal_{W/V}(U)=0$ if the open set $U$ is contained in
$\xg\sm \yg_{\!\hy}$ and $\Qcal_{W/V}(U)=W/V$ otherwise. By
generalizing the proof of Proposition~\ref{prop-sh} one sees that
$H^i(X;\Qcal_{W/V}) \cong H^i(\Gm;W/V)$ for all $i\geq 0$. As a
consequence of the exactness of \eqref{qshe} we obtain a long exact
sequence
\be\label{leq}
\begin{aligned}
\ldots \lra & H^{i-1}(\Gm;W/V)\lra \Hpar^i(\Gm;V,W) \lra
\Hpar^i(\Gm;W)
\\
&\lra H^i(\Gm;W/V) \lra \Hpar^{i+1}(\Gm;V,W)
\lra\ldots
\end{aligned}
\ee



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Maass forms and cohomology}\label{sect-MFsac}
This section generalizes the results concerning the relation between
Maass forms and analytic cohomology given in Chapter~\ref{ch-II} for
cocompact groups to groups with cusps. We follow the same method as
for cocompact groups, with some modifications to handle the
complications caused by the cusps. The presence of cusps brings also
a simplification: The cusps are vertices of the tesselations situated
on $\p\H$. There is no need to extend cocycles to hyperbolic fixed
points on~$\p\H$, like we needed to do in~\S\ref{ssect-injcc}.

\subsubsection{From invariant eigenfunctions to parabolic cocycles}
The linear maps $\bold r$ and $\bold q$ from invariant eigenfunctions
to cohomology classes have been described in \eqref{pqrdef} only in
the group model of cohomology. In a model based on a
tesselation~$\tess$ of type \Fcoh\ or a refinement of it, cocycles
$r$ and~$q$ representing $\bold r u$, respectively $\bold q u$, for
$u\in \E_s^\Gm$, are determined by:
\indind{$\bold r$; $\bold q$}{rqT}
\be \label{rqT}
r(x)(\z) \= \int_x [u,R(\z;\dv)^s]\,,\qquad q(x)(z)\= \int_x [u,
q_s(\dv,z)]
\quad \text{ for }x\in X_1^{\tess,\hy}\,.\ee
We know from Propositions \ref{prop-im-coh} and \ref{prop-mpc-t} that
$\bold r u$ is a parabolic cohomology class in
$\Hpar^1(\Gm;\V\om s,\V\wdg s)$, where $\V\wdg s$ consists of
elements with finitely many singularities of a special type,
discussed in Definition~\ref{wdgdef}. The following result gives
explicit cocycles in the classes $\bold r u$ and~$\bold q u$.
\begin{prop}\label{prop-rqext}Let $\tess$ be a tesselation refining a
tesselation of type~$\Fcoh$.

For a cusp form $u\in \Mf_s^0(\Gm)$, the cocycles $r$ and $q$
in~\eqref{rqT} have unique extensions
$r\in Z^1(\Fpt^\tess;\V\om s,\V\sa s)$ and
$q \in Z^1(\Fpt^\tess;\W\om s,\W\sa s)$, given by the integrals
in~\eqref{rqT} for all $x\in X^\tess_1$.

If $u\in \Mf^1_s(\Gm)$ and $s\neq \frac12$, the cocycle $r$ has a
unique extension as an element of $Z^1(\Fpt^\tess;\V\om s,\V\smp s)$
determined by
\[ r(e_{P,\kp}) \= - \av{\pi_\kp}^+ r(e_{\pi_\kp^{-1}P,P}) \= -
\av{\pi_\kp}^- r(e_{\pi_\kp^{-1}P,P}) \]
for $\kp\in \cu$ and $P\in X_1^\tess$ on the horocycle $V_\hy(\kp)$.
(See \S\ref{ssect-invavpar} for the one-sided averages.)

If $s\neq \frac12$ and $u\in \E_s^\Gm$ is a general invariant
eigenfunction, then $r$ can be extended, non-uniquely, as an element
of $Z^1(\Fpt^\tess;\V\om s,\V\wdg s)$ by defining
\[ r(e_{P,\kp}) \= - \av {\pi_\kp}^+ r(e_{\pi_\kp^{-1}P,P})\]
for $\kp\in \cu$ and for $P\in X^\tess_1$ on the horocycle $V_\hy(\kp)$.

In all cases $q=  \bb(s) \, \tpoiss s r$ gives an extension
$q\in Z^1(\Fpt^\tess;\W\om s,\W\wdg s)$.
\end{prop}
\begin{proof}If $u$ is a cusp form, then the integral in \eqref{rqT}
converges also when one end point of $x$ is a cusp. Use
Proposition~\ref{prop-Vominf} and \eqref{rhh}, and
Proposition~\ref{prop-Tinv0} for the uniqueness. In the general
situation, we have $r(e_{\pi_\kp^{-1}P,P}) \in \V\om s$, and hence
$\av {\pi_\kp}^+ r(e_{P,\pi_\kp^{-1}P}) \in \V\om s[\kp]$.
Lemma~\ref{lem-uupbt} and Proposition~\ref{prop-phavhol} imply, after
 conjugation, that
 $\av {\pi_\kp}^+  r(e_{\pi_\kp^{-1}P,P}) \in \V{\om,\,\wdgwdg} s[\kp]$.
 The $\Gm$-equivariance of $r$ follows from
 $\pi_{\gm^{-1}\kp} = \gm^{-1}\pi_\kp\gm$. For the cocycle property it
 suffices to consider $dr$ on the triangles $V_\kp$ at the cusps
$\kp\in \fdcu$:
\begin{align*}
dr(V_\kp) &\= r(e_{P_\kp,\kp})\,|\,(1-\pi_\kp)+r(e_{\pi_\kp^{-1}P_\kp,P_\kp})
\\
&\= \bigl(- \av {\pi_\kp}^+ r(e_{\pi_\kp^{-1}P_\kp,P_\kp})
\bigr)\,|\,(1-\pi_\kp)
+r(e_{\pi_\kp^{-1}P_\kp,P_\kp})
\=0\,.
\end{align*}

If $u\in \Mf^1_s(\Gm)$, then Proposition~\ref{prop-KH1-smp} implies
that $r(e_{\pi_\kp^{-1}P,P})\in \V\smp s\,|\,(1-\nobreak \pi_\kp)$, and
then the choice $r(e_{P,\kp}) = \av T^+ r(e_{\pi_\kp^{-1}P,P}) 
= \av T^- r(e_{\pi_\kp^{-1}P,P})$ is unique. See Propositions
\ref{prop-Tinv1} and~\ref{prop-kerav-(1)}.
\end{proof}



\subsubsection{From parabolic cocycles to invariant
eigenfunctions}\label{ssect-coie}
The ideas in \S\ref{ssect-coMacc} and \S\ref{ssect-coMaavcc} can be
applied, with some modifications. In this subsection, we construct an
element of $\E_s^\Gm$ starting from a parabolic
$(\W\om s,\W\wdg s)$-valued cocycle in two ways: a local
representation as a sum of values of the cocycle, like in
\S\ref{ssect-coMacc}, and as an average over $\Gm$, like
in~\S\ref{ssect-coMaavcc}.

We work with a tesselation $\tess$ of type \Fcoh\ based on a Dirichlet
fundamental domain~$\fd$. A difference with the cocompact case is the
presence of edges in $X_1^\tess$ with a cusp as one of their end
points. For the interior edges $e\subset\uhp_\hy$ we can work with
$R$-neighborhoods $N_R(e)$, like in \S\ref{ssect-coMacc}. The
$R$-neighborhoods with respect to the hyperbolic distance of edges
$e_{P,\kp}$ with $\kp\in \cu$ intersect infinitely many
$\Gm$-translates of $e_{P,\kp}$. We define instead for
$R>0$\indind{$N_R(\dv)$}{NRcusp}
\be\label{NRcusp}
N_R(e_{P,\kp} ) \= \bigl\{ g_\kp z\;:\; |\re z|\leq R\,,\; \im z\geq
1/R\bigr\}\,. \ee
This is the set $g_\kp W_{1/R}$ with $W_{1/R}$ as defined
in~\eqref{Weps-def}. It is of the form $\uhp\sm\Om$ for an excised
neighborhood $\Om$ of $\proj\RR\sm\{\kp\}$. It contains $e_{P,\kp}$ if
$R\geq\hy$.
\smallskip

For a given cocycle $\ps\in Z^1(\Fpt^\tess;\W\om s,\W\wdg s)$ we
choose a $\Gm$-equivariant lift
$\tilde\ps \in C^1(\Fpt^\tess;\G^\om_s,\G^\wdg_s)$ by first choosing
lifts of $\ps(b)$ for $b$ in a $\QQ[\Gm]$-basis $B$ of $F_1^\tess$;
 e.g., $B=E\cup\{f_\kp,e_\kp\;:\; \kp\in \fdcu\}$. For interior edges
 $e\in E$ or $e=f_\kp$, we choose any lift $\tilde\ps(e)\in \G^\om_s$
of $\ps(e)\in \W\om s$. For the edge $e_\kp$ to the cusp $\kp\in \fdcu$
it is sensible to take a lift of $\ps(e_\kp)\in \W\wdg s$ satisfying
\be\label{lift-cond}
\tilde\ps(e_\kp) \in \G^\wdg_s\,,\qquad\sing\tilde\ps(e_\kp)
\subset N_R(e_\kp)\quad\text{for some }R\geq\hy\,. \ee
To see that this is possible, we note that
$\ps(e_\kp)\,|\,(1-\pi_\kp) = \ps(f_\kp) \in \W\om s$. Hence
$\bsing \ps(e_\kp) \subset\{\kp\}$.

Next we fix $R\geq\hy$ large enough that
$\sing\tilde\ps(e) \subset N_R(e)$ for all $e\in X_1^\tess$. Let $Z$
be a finite union of $\Gm$-translates of~$\fd$. We can find cycles
$C\in \ZZ[X_1^\tess]$ with winding number~$1$ on~$Z$ such that
$N_R(C) = \bigcup_{e\subset\supp C} N_R(e)$ does not intersect~$Z$.
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The cycle $C$ has to pass through the finitely many cusps in~$Z$.

We define $u_\ps$ on~$Z$ by
\be\label{upsdef-cp} u_\ps(z)\= \frac1{\pi i} \tilde\ps(C)(z)\,. \ee
Like in \S\ref{ssect-coMacc} this does not depend on the choice of the
cycle $C$, on the choice of the lift $\tilde\ps$, or on the choice of
$\ps$ in its cohomology class, and satisfies
$u_\ps(\dt z) = u_\ps(z)$, and leads to
$u_\ps = u_{[\ps]}\in \E_s^\Gm$.



 Suppose that $\ps$ is the cocycle $q$ in \eqref{rqT} representing
 $\bold q u$, with $u\in \E_s^\Gm$. We can take a lift
$\tilde\ps(b)\in C^2(\uhp)$ of $q(b)$, for $b\in B$, equal to the
value of the integral in \eqref{rqT} outside $N_\e(b)$. If $\e$ is
sufficiently small, then there is a non-empty open set
$V\subset \mathring\fd_\hy$ not intersecting the $\e$-neighborhood of
any $e\in X_1^\tess$. With Theorem~\ref{thm-psC} and
Proposition~\ref{proppsCi} we obtain for $z\in V$:
\[ u_\ps(z) \= \frac1{\pi i} \int_{\p\fd_\hy} [u,q_s(\dv,z)] \=
u(z)\,.\]
By analyticity $u_\ps=u$ everywhere on~$\uhp$.


If $s\neq \frac12$, we have $\bold q \E_s^\Gm \subset
\Hpar^1(\Gm;\W\om s,\W\wdg s)$, from Theorem~\ref{thm-H1Wow-char}.
Thus, $\ps\mapsto u_\ps$ induces a one-sided inverse of~$\bold q$.
For $s=\frac12$, we have $u_q=u$ only for those $u\in \E_{1/2}^\Gm$
for which $\bold q u \in \Hpar^1(\Gm;\W\om{1/2},\W\wdg{1/2})$. This
includes the Maass cusp forms in $\Mf_{1/2}^0(\Gm)$.

We summarize:
\begin{prop}\label{prop-al-cp}There is a linear map $\ps\mapsto u_\ps$
from $Z^1(\Fpt^\tess;\W\om s,\W\wdg s)$ to $\E^\Gm_s$. The invariant
eigenfunction $u_\ps$ depends only on the parabolic cohomology class
$[\ps]\in \Hpar^1(\Gm;\W\om s,\W\wdg s)$ and can be given on each
compact set in~$\uhp$ by a (finite) $\CC$-linear combination of
translates of $\ps(e)$, where $e$ runs through the $\QQ[\Gm]$-basis
$B$ of $F^\tess_1$ in \eqref{Efd}. If $s\neq \frac12$ the induced map
\indlbl{al-cp1}{$\al_s^\om$}$\al_s^\om: 
\Hpar^1(\Gm;\W\om s,\W\wdg s)\lra \E_s^\Gm$ satisfies
$\al_s^\om\, \bold q=\mathrm{Id} $ on $\E_s^\Gm$.
\end{prop}
In \S\ref{ssect-M0M1} we will discuss the question under what
conditions on $\ps$ the invariant eigenfunction $u_\ps$ is in
$\Mf_s^0(\Gm)$ or $\Mf_s^1(\Gm)$.\medskip

Now we have generalized the approach in \S\ref{ssect-coMacc} leading
to Theorem~\ref{thm-cc-upsi}. To generalize \S\ref{ssect-coMaavcc},
we use the diagram \eqref{PW-diagr}. It satisfies the exactness
condition \eqref{cuex}, as follows from Lemma~\ref{lem-WT0}.
Proposition~\ref{prop-lesmpc} implies that there is a long exact
sequence associated to \eqref{PW-diagr}, from which we use the
following part:
\begin{align*}
 \lra &\Hpar^1(\Gm;\G^\om_s,\G^\wdg_s) \lra \Hpar^1(\Gm;\W\om s,\W\wdg
 s)\\
& \lra \Hpar^2(\Gm;\N^\om,\N^\wdg)
\lra \end{align*}
We choose a lift $\tilde\ps \in C^1(\Fpt^\tess;\G^\om_s,\G^\wdg_s)$ of
$\ps \in Z^1(\Fpt^\tess;\W\om s,\W\wdg s)$ as above. Then
$d\tilde\ps \in Z^2(\Fpt^\tess;\N^\om,\N^\wdg)$. As in
\S\ref{ssect-coMaavcc}, we obtain
\be\label{ups-av} u_\ps \= \frac1{\pi i} \av \Gm\bigl(
d\tilde\ps(\fd)\bigr)\,,\ee
independent of the choice of the lift~$\tilde\ps$. The support of
$d\tilde\ps(\fd)\in \N^\wdg$ is not compact, but meets only finitely
many $\Gm$-translates of~$\fd$. So the sum defining
$\av \Gm\bigl(d\tilde\ps(\fd)\bigr)$ is locally finite and converges
absolutely. The representation of $u_\ps$ as an average does not
depend on the choice of the lift~$\tilde\ps$
satisfying~\eqref{lift-cond}.




\subsubsection{Injectivity} Starting from a cocycle
$\ps \in Z^1(\Fpt^\tess;\W\om s,\W\wdg s)$ and a lift $\tilde\ps$ of
$\ps$ satisfying \eqref{lift-cond}, we have constructed in two ways,
\eqref{upsdef-cp} and \eqref{ups-av}, a $\Gm$-invariant eigenfunction
$u_\ps$, thus obtaining a linear map $\al_s^\om$ from
$\Hpar^1(\Gm;\W\om s,\W\wdg s)$ to $\E^\Gm_s$. Now we will prove:
\begin{prop}\label{prop-i-al}Let $s\in\CC$, $0<\re s<1$. The map
\[\al_s^\om: \Hpar^1(\Gm;\W\om s,\W\wdg s)
\lra \E_s^\Gm\] in Proposition~\ref{prop-al-cp} is injective.
\end{prop}

With Proposition~\ref{prop-al-cp} and with the fact that the
transverse Poisson map gives isomorphisms $\V\om s \cong\W\om s$ and
$\V\wdg s\cong\W\wdg s$, this implies:
\begin{prop}\label{prop-iso-M}
Let $s\in \CC$,\, $0<\re s < 1$, $s\neq\frac12$. Then
\[ \Hpar^1(\Gm;\V\om s,\V\wdg s)
\;\cong \; \E_s^\Gm\,. \]
\end{prop}

To prove Proposition~\ref{prop-i-al} we use the following results, of
which we postpone the proofs.

\begin{prop}\label{prop-H10-cu}The cohomology group
$\Hpar^1(\Gm;\G^\om_s,\G^\wdg_s)$ is zero.
\end{prop}
\begin{lem}\label{lem-av-Nwdg}Suppose
$c\in C^2(\Fpt^\tess;\N^\om,\N^\wdg)$ satisfies the following
conditions:
\begin{enumerate}
\item[i)] There exists $R>0$ such that
$\supp c(V_\kp) \subset N_R(e_\kp)$ for all $\kp\in \fdcu$.
\item[ii)] $\av \Gm\bigl(c(\fd)\bigr)=0$.
\end{enumerate}
Then the class $[c] \in \Hpar^2(\Gm;\N^\om,\N^\wdg)$ is the zero
class.
\end{lem}
\begin{proof}[Proof of Proposition~\ref{prop-i-al}.]For a given
cocycle $\ps\in Z^1(\Fpt^\tess;\W\om s,\W\wdg s)$ we suppose that
$u_\ps=0$. We have to show that $[\ps]=0$ in
$\Hpar^1(\Gm;\W\om s,\W\wdg s)$.

To obtain $u_\ps$ from $\ps$ we have chosen a lift $\tilde\ps \in
C^1(\Fpt^\tess;\G^\om_s,\G^\wdg_s)$ satisfying~\eqref{lift-cond}. The
assumption implies that $ \av \Gm\bigl(
d\tilde\ps(\fd) \bigr) = \pi i\, u_\ps=0$. For each $\kp\in \fdcu$ we
have
\[ d\tilde\ps(V_\kp) \= -
\tilde\ps(f_\kp)+\tilde\ps(e_\kp)|(1-\nobreak\pi_\kp)\]
as an identity in $\G^\wdg_s$. The singularities of $\tilde\ps (f_\K)$
are contained in a compact subset of~$\uhp$, and those of
$\tilde\ps(e_\kp)$, and hence also of
$\tilde\ps(e_\kp)|(1-\nobreak\pi_\kp)$, in a set $N_r(e_\kp)$ as
in~\eqref{NRcusp}. So there is an open neighborhood $\Om$ of
$\proj\RR\sm\{\kp\}$ in~$\proj\CC$ that is an excised neighborhood
of~$\proj\RR\sm\{\kp\}$, for which
$d\tilde\ps(V_\kp) \in \E_s(\Om\cap\uhp)$. Since the cochain
$\tilde\ps$ represents the cocycle $\ps$ we have $d\tilde\ps(V_\kp)=0$
on $\Om \cap\uhp$. In particular
$\supp d\tilde\ps(V_\kp)\subset N_R(e_\kp)$ if $R$ is sufficiently
large. Now we can apply Lemma~\ref{lem-av-Nwdg} to conclude that the
class $[d\tilde\ps]$ in $\Hpar^2(\Gm;\N^\om ,\N^\wdg)$ is zero.

We use the part
\[ \Hpar^1(\Gm;\G^\om_s,\G^\wdg_s) \lra \Hpar^1(\Gm;\W\om s,\W\wdg s)
\stackrel{\dt}\lra \Hpar^2(\Gm;\N^\om,\N^\wdg)\]
of the long exact sequence associated to the diagram~\eqref{PW-diagr}
by Proposition~\ref{prop-lesmpc}. Since $[d\tilde\ps]=\dt[\ps]$ we
have $[\ps]=0$ by Proposition~\ref{prop-H10-cu}.
\end{proof}


\begin{proof}[Proof of Lemma~\ref{lem-av-Nwdg}.] The cocycle $c$ is
determined by its values on the basis
$\{\fd_\hy\}\cup\{V_\kp\;:\; \kp\in \fdcu\}$ of $F_2^\tess$.
\twocolwithpictr{\pridtc The support of $c(\fd_\hy)$ is a compact
subset of~$\uhp$. Condition~i) in the lemma ensures that we can find
a large $a\geq \hy$ such that $\supp c(\fd_\hy) \subset \Gm\, \fd_a$
and that $\supp c(V_\kp) \subset \Gm(\fd_a\cup V_\kp^a)$ for all
$\kp\in \fdcu$. In particular $\supp c(V_\kp)$ does not intersect
$\Gm V^a_\ld$ for some $\ld\neq\kp$, $\ld\in \fdcu$. Condition~ii)
implies that $\av \Gm\bigl(c(V_\kp)\bigr)=0$ on $\Gm \mathring V_\kp^a$.
}{
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We take $\al\in C_c^\infty(\RR)$ such that
$\sum_{n\in \ZZ}\al(x+\nobreak n)=1$ for all $x\in \RR$, and
$\bt\in C^\infty(0,\infty)$ equal to $0$ on $(0,a)$ and equal to~$1$
on $(a+\nobreak \e,\infty)$ for some $\e>0$. We define for each
$\kp\in \fdcu$ the function $\ch_\kp\in \N^\wdg$ by
$\ch_\kp(g_\kp z) = \al(x)\, \bt(y)$. Let $\hat c$ be given by
$\hat c(\fd_\hy)=c(\fd_\hy)$ and for each $\kp\in \fdcu$
\[ \hat c(V_\kp)\= c(V_\kp)- \sum_{n\in \ZZ} \bigl( c(V_\kp)\cdot
(\ch_\kp|\pi_\kp^n)
\bigr)|(1-\pi_\kp^{-n} )\= \ch_\kp \sum_{n\in \ZZ} c(V_\kp)
|\pi_\kp^{-n}\,. \]
The cocycles $\hat c$ and $c$ are in the same cohomology class, and
$\av\Gm\bigl( \hat c(\fd)\bigr)=\av\Gm\bigl(  c(\fd)\bigr)$. Since
the support of $ c(V_\kp)$ is contained in $\Gm_\kp \bigl( V_\kp^a
\cup\nobreak \fd_a\bigr)$ we have for $z\in \mathring V_k^{a+\e}$
\[ \av\Gm\bigl( c(V_\kp) \bigr)(z) \= \ch_\kp(z)\sum_{n\in \ZZ}
c(V_\kp)|\pi_\kp^{-n}(z)
\= \hat c(V_\kp)\,, \]
and on the other hand $\av\Gm\bigl( c(V_\kp)\bigr)(z) = \av\Gm\bigl(
c(\fd_\hy)\bigr)=0$. Since $\e>0$ was arbitrary we conclude that
$\hat c(V_\kp)$ has compact support for each $\kp\in \fdcu$. Hence
$\hat c(\fd) $ has compact support, and is an element of $\N^\om$.


The proof of Proposition~\ref{prop-av-inj} works for $\hat c(\fd)$,
although the support of the function $\ch$ constructed there is not
compact and intersects infinitely many fundamental domains. Thus we
see that $\hat c
(\fd) \in \sum \N^\om |(1-\nobreak\gm)$. The cocycle $\hat c$ is in
the same class as the cocycle $\tilde c$ given by
$\tilde c(\fd_\hy)=\hat c(\fd)$ (which is in
$\sum_\gm \N^\om|(1-\nobreak\gm)$) and $\tilde c(v_\kp)=0$ for
$\kp\in \fdcu$. The cocycle $\tilde c$ is a coboundary.
\end{proof}

\begin{proof}[Proof of Proposition \ref{prop-H10-cu}]Let
$\ps \in Z^1(\Fpt^\tess;\G^\om_s,\G^\wdg_s)$, where $\tess$ is a
tesselation of type~\Fcoh. The presence of cusps gives us directly
$\ps_{P,Q} = \ps(e_{P,Q}) \in \G^\wdg_s$ for all $P,Q\in X_0^\tess$,
including the cusps. There is no need of an extension to hyperbolic
fixed points as in Lemma~\ref{lem-coc-ext}. For
$P,Q\in X_0^{\tess,\hy} = X_0^\tess\cap \uhp$, we have
$\ps_{P,Q}\in \G^\om_s$.

The cocycle $\ps$ is determined by its values on the $\QQ[\Gm]$-basis
$B= E \cup \{f_\kp\}\cup\{e_\kp\}$ in~\eqref{Efd}. For interior edges
we have $\ps(e), \ps(f_\kp) \in \G^\om_s$. Since $\ps$ is a cocycle,
we have $\ps(e_\kp)
\,|\,(1-\nobreak \pi_\kp) = \ps(f_\kp)$. This implies that
\eqref{lift-cond} holds automatically for the cocycle~$\ps$. Thus, we
know that there is a number $R>0$ such that
$\sing \ps(x) \subset N_R(x)$ for all $x\in X_1^\tess$, with
$N_R(\cdot)$ as defined in~\S\ref{ssect-coie}.


Let cusps $\x,\eta\in \cu$ be given, and let $Z\subset \uhp$ be
compact. There is a chain $p= \sum_j \e_j\, e_j \in \ZZ[X_1^\tess]$,
representing a path from $\x$ to $\eta$, with $\e_j\in \{1,-1\}$,
$e_j\in X_1^\tess$ such that $N_R(e_j) \cap Z = \emptyset$ for
all~$j$. For $z\in Z$ we have
\[ \ps_{\x,\eta} (z) \= \ps(p)(z) \= \sum_j \e_j \,\ps(e_j)( z)\,.\]
So $\ps_{\x,\eta}\in \E_s\bigl( \mathring Z \bigr)$. The path can be
adapted to any compact $Z\subset\uhp$, and hence
$\ps_{\x,\eta}\in \E_s$ for each pair of cusps.

Let $P\in X_0^{\tess,\hy}$. {}From
$\ps_{P,\x} \,|\,(1-\nobreak\pi_\x) = \ps_{P,\pi_\x^{-1}P} \in \G^\om_s$
it follows that
$\overline{\sing(\ps_{P,\x})} \cap\p\uhp \subset \{\x\}$. Let
$\eta\in \cu$, $\eta\neq \x$. In the relation
$\ps_{\x,\eta} = \ps_{P,\eta}-\ps_{P,\x}$ singularities near~$\x$ of
$\ps_{P,\x}$ cannot be canceled by singularities of $\ps_{P,\eta}$.
So $\sing\bigl(\ps_{P,\x}\bigr)$ is a compact set in~$\uhp$.

We apply Lemma~\ref{lem-Pow-Po} with $\hat q=\ps_{P,\x}\,|\,g_\x$.
Since
$\ps_{P,\x}\,|\,(1-\nobreak\pi_\x) = \ps_{P,\pi_\x^{-1}P}\in\G^\om_s$,
condition~b) is satisfied as well. So there is
$p_\x\in\G^\wdg_s \cap \E_s^{\pi_\x}$ such that
$\ps_{P,\x}- p_\x\in\G^\om_s$. For $\eta = \gm^{-1}\x$, $\gm\in\Gm$,
we take $p_\eta=p_\x\,|\,\gm$. The $\pi_\x$-invariance of~$p_\x$
implies that this does not depend on the choice of $\gm$ such that
$\eta=\gm^{-1}\x$.

Let $h\in C^0(\Fpt^\tess;\G^\om_s,\G^\wdg_s)$ be given by $h(Q)=0$ if
$Q\in X_0^\tess\cap\uhp$, and by $h(\x)=p_\x$ for $\x\in\cu$. So $h$
takes values in $\E_s$. We go over to the cocycle $\hat\ps=\ps-dh$ in
the cohomology class of~$\ps$. Then $\hat \ps_{\x,\eta} \in \E_s$ for
all $\x,\eta\in \cu$, and
$\hat\ps_{P,\x}= \ps_{P,\x} - p_\x\in \G^\om_s$. So
$\hat\ps_{\x,\eta} = \hat \ps_{P,\eta}-\hat\ps_{P,\x}
\in\E_s\cap \G^\om_s = \{0\}$, as follows from~\eqref{F-decomp}.
Taking the base point in $\cu$ we obtain a group cocycle
corresponding to $\hat\ps$ that is zero.
\end{proof}

\subsubsection{Restriction to subspaces}\label{ssect-M0M1}
The map $\al_s^\om:\Hpar^1(\Gm;\W\om s,\W\wdg s)\lra \E_s^\Gm$ has
been constructed in \S\ref{ssect-coie} for all $s$ with $0<\re s<1$.
Under the additional condition $s\neq \frac12$ it is an isomorphism.
Now we turn to its restriction to subspaces of
$\Hpar^1(\Gm;\W\om s,\W\wdg s)$.

\begin{prop}\label{prop-M0M1}Let $s\in \CC$, $0<\re s<1$. The
following linear maps are isomorphisms:
\be\label{Pal-sa}
\Hpar^1(\Gm;\V\om s,\V\sa s)
\stackrel{\tpoiss s}\lra \Hpar^1(\Gm;\W\om s,\W\sa s)
\stackrel{\al_s^\om}\lra \Mf^0_s(\Gm)\,.
\ee
Under the additional condition $s\neq \frac12$, the following linear
maps are isomorphisms as well:
\be\label{Pal-smp}
\Hpar^1(\Gm;\V\om s,\V\smp s)
\stackrel{\tpoiss s}\lra \Hpar^1(\Gm;\W\om s,\W\smp s)
\stackrel{\al_s^\om}\lra \Mf^1_s(\Gm)\,. \ee
\end{prop}
\begin{proof}We consider two cases: (a)
$W=\W\sa s$, $s=\frac12$ allowed; (b)
$W=\W\smp s$, $s\neq \frac12$. Consider a group cocycle
$\ps\in \Zpar^1(\Gm;\W\om s,W)$. Proposition~\ref{prop-al-cp} implies
that $[\ps] = \bold q u_\ps$.

Consider a cusp $\kp\in \fdcu$. Propositions \ref{prop-KH1-sa}
and~\ref{prop-KH1-smp} imply that there is a unique eigenfunction
$v_\kp $ such that the class $\bold q v_\kp \in H^1(\Gm_\kp;\W\om s,W)$
is represented by the restriction of $\ps$ to~$\Gm_\kp$, {\sl i.e.},
$\ps_{\pi_\kp}=v_\kp$. We have $v_\kp \in (\K^0_s)^\uparrow$ in case
(a), and $v_\kp\in \K^\uparrow_s$ in case~(b).
Proposition~\ref{prop-rinj} shows that $v_\kp=u_\ps$. Since this holds
for all cusps $\kp\in \fdcu$, we conclude that
$u_\ps \in \Mf^0_s(\Gm) $ in case (a), and $u_\ps\in \Mf^1_s(\Gm)$ in
case~(b). Proposition~\ref{prop-im-coh} shows that $\al_s^\om$ gives
isomorphisms in \eqref{Pal-sa} and~\eqref{Pal-smp}. The proof is
completed by the fact that the transverse Poisson map gives
isomorphisms $\V\om s \rightarrow\W\om s$ and
$\V\wdg s\rightarrow\W\wdg s$.
\end{proof}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Parabolic cohomology and mixed parabolic
cohomology}\label{sect-pcmpc}
In this section we shall prove the isomorphism
\beqabclbl{pcm}
\be\label{pcm-a} \Hpar^1(\Gm;\V\om s,\V\sa s)
\;\cong\; \Hpar^1(\Gm;\V\sa s) \ee
in Theorem~\ref{thmB}, and also, under the assumption $s\neq\frac12$,
the isomorphisms
\begin{align}
\label{pcm-b}
\Hpar^1(\Gm;\V\om s,\V\smp s) &\;\cong\; \Hpar^1(\Gm;\V\smp s)\,,\\
\label{pcm-c}
\Hpar^1(\Gm;\V\om s,\V\wdg s) &\;\cong\; \Hpar^1(\Gm; \V\wdgcp s)\,.
\end{align}
\eeqabc
Together with the previous section, this will complete the proof of
Theorem~\ref{thmC}.

We recall that $\V\wdgcp s$ consists of the $f\in \V\wdg s$ for which
the set of singularities $\bsing (f) $ is contained in the set $\cu$
of cusps of~$\Gm$. In Proposition~\ref{prop-commut} we shall give an
example that shows that there are $\Gm$ for which we cannot replace
$ \V\wdgcp s$ by $\V\wdg s$ in~\eqref{pcm-c}. The proofs will show
that \eqref{pcm-c} still holds if we replace $\V\wdgcp s$ by the
$\Gm$-module of those $f\in \V\wdg s$ for which $\bsing( f)$ does not
contain hyperbolic fixed points.



\subsubsection{Space of singularities}The $\QQ[G]$-module
\indlbl{singq}{$\Scal_s$; $\Scal_{s,\x}$}$\Scal_s = \V\fs s/\V\om s$
is the space of singularities of semi-analytic vectors in the
principal series. For $\x\in \p\uhp$ we denote by $\Scal_{s,\x}$ the
subspace represented by elements of $ \V\om s[\x]$. For each $g\in G$
the map $f\mapsto f\,|\,g$ induces an isomorphism
\[ \Scal_{s,\x} \lra \Scal_{s,\x}\,|\,g \= \Scal_{s, g^{-1}\x}\,. \]
Clearly, $\Scal_s$ contains the direct sum of all $\Scal_{s,\x}$ with
$\x\in \proj\RR=\p\uhp$. We note that $\Scal_{s,\x}$ is a subspace of
$\Scal_s$, not a stalk of a sheaf.
\begin{prop}\label{prop-sing-dec}The space $ \Scal_s $ is equal to
$\displaystyle \bigoplus_{\x\in \proj\RR} \Scal_{s,\x}$.
\end{prop}
\begin{proof}Suppose that $f\in \V\fs s$ represents an element of
$\Scal_s$. We write $\bsing (f) = \{\x_1,\ldots,\x_n\}$. We identify
$f$ (in the circle model) with a holomorphic function
$f\in \hol(\Om)$, where $\Om \subset\CC$ is an open set such that
$\uc \sm \Om = \{\x_1,\ldots,\x_n\}$. We choose open sets $\Om_1$ and
$\Om_2$ such that
\begin{align*}
& \Om \= \Om_1\cap \Om_2\,,\\
& \x_1 \in \Om_1\,,\; \x_2,\ldots,\x_n \not \in \Om_1\,,\qquad
\x_1\not\in \Om_2\,,\; \x_2,\ldots,\x_n \in \Om_2\,.
 \end{align*}
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A theorem in complex function theory (e.g., \cite{Hor}, Proposition
1.4.5) gives the existence of $f_1 \in \hol(\Om_1)$,
$f_2\in \hol(\Om_2)$, such that $f=f_1+f_2 $ on~$\Om$.
(Apply Theorem~1.45 in~\cite{Hor} to the covering $\{\Om_1,\Om_2\}$ of
$\Om_1\cup\Om_2$, and put $g_{1,2}=-g_{2,1} = f$ on $\Om_1\cap\Om_2$.
The theorem gives $g_j \in \hol(\Om_j)$ such that $g_{1,2}=g_1-g_2$
on $\Om_1\cap \Om_2$.)
Thus we have $\bsing (f_1) \subset \{\x_1\}$ and
$\bsing(f_2 )\subset \{\x_2,\ldots,\x_n\}$. Repeating this
construction gives $f+\V\om s$ as an element of
$\bigoplus_{\x\in \proj\RR} \Scal_{s,\x}$.
\end{proof}


\sssect{Exact sequence} For any $\QQ[\Gm]$-module~$W$ with
$\V\om s \subset W \subset\V\fs s$, we conclude from \eqref{leq} that
the following sequence is exact
\be H^0(\Gm; W/\V\om s) \lra \Hpar^1(\Gm;\V\om s,W) \lra
\Hpar^1(\Gm;W)
\lra H^1(\Gm;W/\V\om s)\,. \ee
Since all $\Gm$-orbits in $\p\uhp$ are infinite, we have
$(W/\V\om s)^\Gm \subset \Scal_s^\Gm = \{0\}$. Hence the natural map
$\Hpar^1(\Gm;\V\om s,W) \lra \Hpar^1(\Gm;W)$ is injective. If the
image of $\Hpar^1(\Gm;W)$ in $H^1(\Gm;W/\V\om s)$ is zero, then this
map is surjective as well.

It seems unnatural that we go from parabolic cohomology to standard
cohomology. The following lemma makes this step more explicit in the
description of cohomology based on a tesselation $\tess$ of
type~\Fcoh, discussed in \S\ref{ssect-res-c}.

Recall that $X_1^\tess$ has three kind of edges: the edges going to a
cusp, which are the $\Gm$-translates of finitely many $e_\kp$
($\kp\in \fdcu$), the edges in $\p\uhp_\hy$, which are the
$\Gm$-translates of finitely many $f_\kp \in X^{\tess,\hy}_1$
($\kp\in \fdcu$), and the interior edges in $X^{\tess,\hy}_1 $ of the
form $\gm\,e$ with $\gm\in\Gm$, $e\in E$.
\begin{lem}\label{lem-cocbd}Let $c_1\in Z^1(\Fpt^\tess;W)$. There is a
cocycle $c$ in the same cohomology class such that $c(e)=0$ for all
edges~$e$ going to a cusp and for all edges contained in
$\p\uhp_\hy$.
\end{lem}
\begin{proof}For each $\kp \in \fdcu$, the edge $e_\kp$ goes from a
point $P_\kp\in \p\uhp_\hy$ to~$\kp$. Define
$f\in \map(X^\tess_0,W)^\Gm$ by $f(P_\kp)=c_1(e_\kp)$ for all
$\kp\in \fdcu$, and $f=0$ on all other $\Gm$-orbits in~$X_0^\tess$.
Take $c=c_1-df$, then $c(e_\kp)=0$ and
$c(f_\kp) = c(e_\kp)\,|\,(1-\nobreak\pi_\kp)=0$ for all $\kp\in \fdcu$.
\end{proof}
The new cocycle $c$ is effectively a $1$-cocycle on
$\Fpt^{\tess,\hy}$, and this resolution computes the standard group
cohomology, as we have discussed in~\S\ref{ssect-Tcoh}. Actually, the
condition that $c(f_\kp)=0$ for all $\kp\in \fdcu$ for some $c$ in a
cohomology class can be used to characterize $\Hpar^1(\Gm;W)$ inside
$H^1(\Gm;W)$.


\begin{defn}\label{def-locdef}We call a $\QQ[\Gm]$-module~$W$ such
that $\V\om s\subset W \subset\V\fs s$ \emph{locally defined} if the
image $\Scal^W_s$ of $W$ in $\Scal_s$ has the form\indind{locally
defined module; $\Scal^W_s$}{def-locdef}
\[ \Scal^W_s \= \bigoplus_{\x\in \proj\RR} \Scal_{s,\x}^W\,,\]
where
\indlbl{SlocW}{$\Scal_{s,\x}^W$}$\Scal_{s,\x}^W 
= \Scal^W_s \cap \Scal_{s,\x}$.
\end{defn}
Since $W$ is a $\QQ[\Gm]$-module, the local summands satisfy
$\Scal_{s,\x}^W \,|\, \gm = \Scal_{s,\gm^{-1}\x}^W$ for all
$\x\in \proj\RR$. The subspaces $\V\sa s$, $\V\smp s$, $\V\wdgcp s$
and $\V\wdg s$ of $\V\fs s$ are all locally defined.


If $W$ is a locally defined $\QQ[\Gm]$-module between $\V\om s$ and
$\V\fs s$, we have
\be H^1(\Gm;\Scal_s^W) \cong \bigoplus_{x\in \Gm\backslash \proj\RR}
H^1\bigl(\Gm;\Scal^W_s(x) \bigr)\,, \ee
where $\Scal^W_s(x)$ is the $\QQ[\Gm]$-module
$\bigoplus_{\x\in x} \Scal_{s,\x}^W$. So for the bijectivity of the
natural map $\Hpar^1(\Gm;\V\om s,W) \lra \Hpar^1(\Gm;W)$ it suffices
to show that the image of
\[ \Hpar^1(\Gm;W) \lra H^1\bigl(\Gm;\Scal_s^W(x)\bigr)\]
is zero for all $\Gm$-orbits $x\in \Gm\backslash\proj\RR$.

The main result of this subsection is the following proposition:
\begin{prop}\label{prop-W}Let $W$ be a locally defined
$\QQ[\Gm]$-module between $\V\om s$ and $\V\fs s$. Let
$\x_0\in \proj\RR$. If $\x_0$ is a hyperbolic fixed point fixed by
$\eta\in\Gm$, $\eta\neq 1$, we assume that the map
\be \label{hypmap}
1-\eta : \Scal^W_{s,\x_0} \lra \Scal_{s,\x_0}^W \ee
is injective. Then the image of $\Hpar^1(\Gm;W) $ in
$H^1\bigl(\Gm;\Scal^W_s(\Gm\x_0)\bigr)$ vanishes.
\end{prop}
This implies that $\Hpar^1(\Gm;\V\om s,W) \cong \Hpar^1(\Gm;W)$ for
all locally defined $W$ between $\V\om s$ and $\V\fs s$ for which the
map in \eqref{hypmap} is injective for all hyperbolic fixed points
of~$\Gm$.

\begin{proof}The proof is long. Starting from
 $c\in Z^1\bigl(\Fpt^{\tess,\hy};\Scal_s^W(\Gm\x_0)\bigr)$
representing a class in the image of $\Hpar^1(\Gm;W)$, we will show
that $c$ is a coboundary, separating the cases where $\Gm_{\x_0}$ is
trivial, hyperbolic or parabolic.

We can assume by Lemma~\ref{lem-cocbd} that $c(f_\kp)=0$ for each
$\kp\in \fdcu$, and hence $c(f)=0$ for any edge $f\in X_1^{\tess,\hy}$
with support contained in $\p\uhp_\hy$.


Let $\x_0\in \proj\RR$. For $h \in \Scal^W_s(\Gm\x_0)$ and
$\x\in \Gm\x_0$ we denote by $h_\x$ the component of~$h$ in the
summand $\Scal_{s,\x}^W$. We have
\be \label{gas}
\bigl( h\,|\,\gm\bigr)_\x \= h_{\gm\x}\,|\,\gm\,. \ee


We put for $\x\in \Gm\x_0$:
\be D(\x) \= \{ e \in X_1^{\tess,\hy}\;:\; c(e)_\x\neq0\}\,. \ee
\begin{lem}For each $\x\in \Gm\x_0$ the set $D(\x)$ consists of
finitely many $\Gm_\x$-orbits.
\end{lem}
\begin{proof}From \eqref{gas}:
\[ \gm e \in D(\x) \Longleftrightarrow c(\gm e)_\x =
c(e)_{\gm^{-1}\x}\,|\,\gm^{-1}\neq0\Longleftrightarrow
c(e)_{\gm^{-1}\x}\neq0\Longleftrightarrow e\in D(\gm^{-1}\x)\,. \]
This implies that $\Gm_\x D(\x) = D(\x)$.

For each interior edge $e\in X_1^{\tess,\hy}$, the set
$\{\x\in \Gm\x_0\;:\; c(e)_{\x}\neq0\}$ is finite, since $\bsing(f)$
is finite for each $f\in \V\fs s$. We use that $X_1^{\tess,\hy}=
\bigsqcup_{e\in E} \Gm \, e \sqcup \bigsqcup_{\kp\in \fd^\cu} \Gm\, f_\kp$,
where $E\cup\{ f_\kp\}$ is the finite $\QQ[\Gm]$-basis of
$F_1^{\tess,\hy}$ mentioned in \S\ref{ssect-res-c}. Since
$c(f_\kp)=0$, we have
\[ D(\x) \= \bigcup_{e\in E}\; \{\gm e\;:\; \gm\in \Gm\,,\;
c(e)_{\gm^{-1}\x}\neq0\}\,, \]
which consists of finitely many $\Gm_\x$-orbits.
\end{proof}

\sssect{Case $\Gm_{\x_0}=\{1\}$}
\twocolwithpictr{\pridtc Let $\kp\in \fdcu$ , and $\gm\in \Gm$. If
$\gm\in \Gm_\kp$, then $c(\gm^{-1}P_\kp,P_\kp)=0$, since there is a path
from $\gm^{-1}P_\kp$ to $P_\kp$ along the horocycle of $P_\kp$. If
$\gm\not\in \Gm_\kp$, then $\gm^{-1}P_\kp$ is on another horocycle. For
each $\x\in \Gm\x_0$ the set $D(\x)$ is finite. Hence there is a path
$p$ from $\gm^{-1}P_\kp$ to $P_\kp$ along edges of $X_1^{\tess,\hy}$
none of which is in $D(\x)$. (It does not matter if the path goes
through an end point of an edge in $D(\x)$.) Hence
$c(\gm^{-1}P_\kp,P_\kp)_\x= c(p)_\x=0$. This holds for all
$\x\in \Gm\x_0$, so $c(\gm^{-1}P_\kp,P_\kp)=0$.}{
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Take $P_\kp$ as the base point for the transition to group cocycles.
This leads to the cocycle $\ps$ satisfying
\[ \ps_\gm \= c(\gm^{-1}P_\kp,P_\kp) = 0\qquad
(\gm\in \Gm)\,.\]
This shows that $c$ represents the trivial cohomology class.


\sssect{Case $\x_0\in \cu$}
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}{\pridtc We take $\kp=\x_0$, and use $P_\kp$ as the base point. As
before, we have $c(\gm^{-1}P_\kp,P_\kp)=0$ if $\gm\in \Gm_\kp$. Let
$\gm\in \Gm \sm \Gm_\kp$. For each $\x\in \Gm\,\kp$ the edges in the
$\Gm_\x$-invariant set $D(\x)$ are all contained in a horocyclic
disk~$C$ at~$\x$. If $\x\neq\kp$ and $\x\neq \gm^{-1}\kp$, a path from
$\gm^{-1}P_\kp$ to $P_\kp$ along edges of $X_1^{\tess,\hy}$ may be
forced to pass through~$C$, but if so, the
pieces~of~the\break}\vskip-1.7ex
\noindent
path that are inside $C$ can be chosen along edges in $\p\uhp_\hy$, on
which $c$ vanishes. Thus we conclude that $c(\gm^{-1}P_\kp,P_\kp)_\x=0$
for $\x\not\in \{\kp,\gm^{-1}\kp\}$.

The group cocycle $\ps_\gm = c(\gm^{-1}P_\kp,P_\kp)$ is of the form
\be \ps_\gm \= (\ps_\gm)_{\gm^{-1}\kp} +
(\ps_\gm)_\kp \in \Scal_{s,\gm^{-1}\kp}^W \oplus \Scal_{s,\kp}^W\,. \ee
Let $\gm,\dt\in \Gm$ such that $\dt\not \in \Gm_\kp$ and
$\gm\dt\not\in \Gm_\kp$. Writing out the cocycle relation
$\ps_{\gm\dt} = \ps_\gm\,|\,\dt+\ps_\dt$, we find for the components
in $\Scal_{s,\kp}^W$:
\be (\ps_{\gm\dt})_\kp \= (\ps_\dt)_\kp\,.\ee
This implies that there is $b_\kp \in \Scal_{s,\kp}^W$ such that
$(\ps_\gm)_\kp = b_\kp$ for all $\gm\in \Gm\sm\Gm_\kp$. For such~$\gm$:
\[ (\ps_{\gm^{-1}})_{\gm\kp} + b_\kp \= \ps_{\gm^{-1} } \= -
\ps_\gm\,|\,\gm^{-1}\=
-(\ps_\gm)_{\gm^{-1}\kp}\,|\,\gm^{-1} - b_\kp\,|\,\gm^{-1}\,.\]
Hence
\[
(\ps_\gm)_{\gm^{-1}\kp} \= -
(\ps_{\gm^{-1}})_{\gm \kp}\,|\,\gm - b_\kp\,|\,\gm - b_\kp
\stackrel{\eqref{gas}}{\=} - \bigl(\ps_{\gm^{-1}} \,|\,\gm \bigr)_\kp
- b_\kp\,|\,\gm- (\ps_\gm)_\kp \= - b_\kp\,|\,\gm\,. \]
We arrive at
\be \ps_\gm \= \begin{cases} 0&\text{ if }\gm\in \Gm_\kp\,,\\
b_\kp\,|\,(1-\gm)&\text{ if }\gm\in \Gm\sm\Gm_\kp\,.\end{cases}
\ee
For $\dt\in \Gm_\kp$, $\gm\in \Gm\sm\Gm_\kp$, the cocycle relation
implies $\ps_{\dt\gm}=\ps_\gm$. Hence
$b_\kp\,|\,(1-\nobreak\dt\gm) = b_\kp\,|\,(1-\nobreak\gm)$, so
$b_\kp \in
(\Scal_{s,\kp}^W)^{\Gm_\kp}$. Thus we have
$\ps_\gm = b_\kp\,|\,(1-\nobreak\gm)$ for all $\gm\in \Gm$, and $c$
represents the trivial cohomology class.

\sssect{Case where $\x_0$ is a hyperbolic fixed point}
\twocolwithpictr{\pridtc We now suppose that $\x_0$ is fixed by a
hyperbolic $\eta\in \Gm$. We fix $\kp\in \fdcu$. Proceeding as in the
previous cases we find $c(\gm^{-1}P_\kp,P_\kp)=0$ if $\gm\in \Gm_\kp$.

\pridtc Let $\gm \in\Gm\sm\Gm_\kp$ and consider $\x\in \Gm\x_0$. Let
$\eta\in \Gm$ be a generator of $\Gm_\x$; say $\x$ is the attracting
fixed point $\om(\eta)$ of~$\eta$. Then $\eta$ also generates
$\Gm_{\x'}$, where $\x'=\al(\eta)$ is the repelling fixed point.} {
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\put(1.7,-1.6){\circle*{.08}}
\put(-1.05,2.4){$\kp$}
\put(-.6,1.8){$P_\kp$}
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\put(1.5,-2.3){$\dt^{-1}\kp$}
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\put(.9,2.4){$\x$}
\put(-1.768,-1.768){\circle*{.08}}
\put(-2.1,-2){$\x'$}
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The set $D(\x)$ consists of finitely many $\Gm_\x$-orbits. If $\kp$ and
$\gm^{-1}\kp$ are in the same cyclic interval
$(\x,\x')_c \subset\proj\RR$ or $(\x',\x)_c\subset\proj\RR$, then we
can find a path from $\gm^{-1}P_\kp$ to $P_\kp$ not containing edges in
$D(\x)$, and hence $c(\gm^{-1}P_\kp,P_\kp)_\x=0$. If $\dt^{-1}\kp$ and
$\kp$ are separated by $\x$ and $\x'$ in $\proj\RR$, then $D(\x)$ may
form a barrier between $\dt^{-1}P_\kp$ and $P_\kp$. But now we have
$c(P_\kp,\eta^{-1}P_\kp)_\x=0$ and
$c(\dt^{-1}P_\kp,\eta^{-1}\dt^{-1}P_\kp)_\x=0$, since $\kp$ and
$\eta^{-1}\kp$ are not separated by $\x$ and $\x'$, and similarly for
$\dt^{-1}\kp$ and $\eta^{-1}\dt^{-1}\kp$. Thus we have
\[ \bigl(c(\dt^{-1}P_\kp,P_\kp)\,|\,(1-\eta)\bigr)_\x \=
c(\dt^{-1}P_\kp,\eta^{-1}\dt^{-1}P_\kp)_\x-
c(P_\kp,\eta^{-1}P_\kp)_\x\=0\,.\]
The injectivity of the map in \eqref{hypmap} implies by conjugation
that $1-\eta$ is injective on $\Scal^W_{s,\x}$, and hence
$c(\dt^{-1}P_\kp,P_\kp)_\x=0$. We proceed as in the case
$\Gm_{\x_0}=\{1\}$.
\medskip

This completes the proof of Proposition~\ref{prop-W}.
\end{proof}



\begin{prop}\label{prop-conds}For the spaces $\V\sa s$, $\V\smp s$ and
$\V\wdgcp s$ the map \eqref{hypmap} in Proposition~\ref{prop-W} is
injective for all hyperbolic fixed points.
\end{prop}
\begin{proof}Suppose after conjugation that
$\eta=\matc{\sqrt t}00{1/\sqrt t}$ with $0<t<1$, fixing~$0$. If
$f\in \V\sa s$ then $f$ has an asymptotic behavior
$f(x) \sim \sum_{n=0}^\infty a_n \,  x^n$ as $x\rightarrow0$ for some
$a_n\in \CC$. Then
$f\,|\,(1-\nobreak\eta)\,(x) = f(x) - t^sf(tx) \sim
\sum_{n=0}^\infty (1-\nobreak t^{n+s}) \,   a_n \,   x^n$. If
$f\,|\,(1-\nobreak\eta)\in \V\om s$, then
$(1-\nobreak t^{n+s}) \,   a_n = \oh(C^n)$ for some $C>0$, and
consequently $a_n$ is also $\oh(C^n)$, since $1-t^{n+s}$ tends to~$1$
for $n$ large, so $f\in \V\om s$. This shows that $\V\sa s$ satisfies
the condition.

If $f\in \V\smp s$, then $f(x) \sim \sum_{n=-1}^\infty a_n x^n$. Now
the assumption that $f\,|\,(1-\nobreak\eta)$ is analytic at $x=0$
implies $(a-\nobreak t^{s-1})a_{-1}=0$, and hence $a_{-1}=0$ since
$\re s\neq1$. So $f$ is analytic at~$0$ as before.

Finally, of course, the condition is vacuous for~$\V\wdgcp s$.
\end{proof}

For the space $\V\wdg s$ the map in \eqref{hypmap} is not injective.
This one sees by considering $\eta=\matc t00{1/t}$ with $t>0$. The
function given by
\be\label{hypex}
\ph(z)\=z^{-s}\quad\text{ for }\re z>0\,,\qquad \ph(z)\=0\quad\text{
for } \re z<0\ee
determines a non-zero $\eta$-invariant element of $\V\wdg s$
(line model), with singularities $0$ and $\infty$. However, of course,
the subspace of those $f\in \V\wdg s$ for which $\bsing( f)$ does not
contain hyperbolic fixed points also satisfies the conditions of
Proposition~\ref{prop-W}.



\sssect{Counterexample} To show that the injectivity of the map in
\eqref{hypmap} is necessary, we give a counterexample, based on the
\emph{commutator subgroup}
\indlbl{Gcom}{commutator subgroup; $\Gcom$}$\Gcom=[\Gmod,\Gmod]$ of
the modular group. It is a subgroup of~$\Gmod$ of index~$6$. It is
free on the hyperbolic generators $C=\matc2111$ and
$D=\matr2{-1}{-1}1$. It has one cuspidal orbit $\proj\QQ$, and
$\left( \Gcom\right)_\infty$ is generated by $T^6=CDC^{-1}D^{-1}$.
See \cite{Leh}, Chap.~XI, \S3E, on p.~362.

\begin{prop}\label{prop-commut}Denote $\gr=\frac{1+\sqrt 5}2$. Then
$\Hpar^1(\Gcom;\V\wdg s)$ has non-zero image in
$H^1\bigl(\Gcom;\Scal_s^W(\Gcom\gr)\bigr)$, where $W$ denotes
$\V\wdg s$.
\end{prop}
\begin{proof}The element $D$ is conjugate to
$\eta= \matc {\gr^2}00{\gr^{-2}}$, and fixes $-\gr$ and $\gr^{-1}$.
The example in \eqref{hypex} shows that we can find $f\in W^D$ with
$\bsing (f) = \{
-\gr,\gr^{-1}\}$.

Define $\ps \in Z^1(\Gcom;W)$ in the group model of cohomology by
$\ps _C=f$ and $\ps_D=0$. This determines a parabolic cocycle:
\[ \ps_{T^6} \= \bigl(\ps_C|_{2s}\,(D-1) + \ps_D|_{2s}\,(1-C)\bigr)
\bigr|_{2s}\, C^{-1}D^{-1} \= f|_{2s}\,(D-1)C^{-1}D^{-1}+0\= 0 \,.\]

Suppose that the image of $[\ps]$ in $H^1(\Gcom;\Scal_s^W(\Gm\gr)$ is
zero. Then the image of $\ps$ is of the form $dF$ with
$F \in \Scal_s^W(\Gm\gr)$. {}From $F\,|\,(D-\nobreak1)=0$ it follows
that $F=F_{-\gr}+F_{\gr^{-1}} \in
\Scal^W_{s,-\gr}  \oplus \Scal^W_{s,\gr^{-1}}$, since $-\gr$ and
$\gr^{-1}$ are the fixed points of~$D$. Further,
$f+\V\om s = F\,|\,(C-\nobreak1) = (F\,|\,C)_{C^{-1}(-\gr)} +
(F\,|\,C)_{C^{-1}\,\gr^{-1}} - F_{-\gr} - F_{\gr^{-1}}$. The points
$C^{-1}(-\gr)$, $C^{-1}\,\gr^{-1}$, $-\gr$ and $\gr^{-1}$ are all
different. Since $\bsing(f) = \{-\gr,\gr^{-1}\}$, we conclude that
$0=(F\,|\,C)_{C^{-1}(-\gr)} = F_{-\gr}\,|\,C$ and
$0=(F\,|\,C)_{C^{-1}\gr^{-1}} = F_{\gr^{-1}}\,|\,C$. Hence
$F=F_{-\gr} + F_{\gr^{-1}}$ vanishes, a contradiction.\end{proof}


\subsubsection{Recapitulation of the proof of
Theorem~\ref{thmC}}\label{recapC}
Let $s\neq\frac12$. The injective map \[ \bold r : \E_s^\Gm \lra
H^1(\Gm;\V\om s)\]
in Proposition~\ref{prop-rinj} has image in
$\Hpar^1(\Gm;\V\om s,\V\wdgcp s) 
\cong \Hpar^1(\Gm;\V\om s,\V\wdg s)$, according to
Proposition~\ref{prop-im-coh}. Proposition~\ref{prop-iso-M} shows
that it is an isomorphism. The space $\V\wdgcp s$ is locally defined
and satisfies the conditions of Proposition~\ref{prop-W} according to
Proposition~\ref{prop-conds}. So
$\Hpar^1(\Gm;\V\om s,\V\wdgcp s) \cong \Hpar^1(\Gm;\V\wdgcp s)$ by
Proposition \ref{prop-W}.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Period functions and periodlike functions for the full
modular group}\label{sect-pf}
We return to the \indind{modular group}{sect-pf}modular group
$\Gmod=\PSL_2(\ZZ)$, which was the sole discrete subgroup of
$\PSL_2(\RR)$ considered in the earlier paper~\cite{LZ}. We give a
cohomological interpretation of the period functions and the
periodlike functions considered there. We show that the cohomology
group $H^1(\Gmod;\V\om s)$ is larger than the image
$\bold r\E_s^{\Gmod}$, and end by describing briefly the
generalization of $\Gm$-invariant eigenfunctions corresponding to the
classes in this larger cohomology group: the quantum Maass forms.

\subsubsection{Periodlike functions and cocycles} The space
$\pf_s(\CC')$ of
\indlbl{plf}{periodlike function; $\pf_s(\dv)$;
$\CC'=\CC\sm(-\infty,0]$}\emph{periodlike functions} on
$\CC'=\CC\sm(-\infty,0]$ is defined, in Chap.~III of~\cite{LZ}, as
the space of functions $\ps:\CC'\lra\CC$ that satisfy the
\indind{three term equation}{3TC}three term equation on~$\CC'$:
\be\label{3TC}
\ps(\tau) \= \ps(\tau+1) + (\tau+1)^{-2s} \,
\ps\Bigl(\frac\tau{\tau+1}\Bigr)\,. \ee
The subspace of holomorphic functions in $\pf_s(\CC')$ is denoted
\indlbl{pfss}{$\pf_s(\dv)_\om$, $\pf_s(\dv)_\infty$}$\pf_s(\CC')_\om$.
Similarly, the space of functions on $\RR_+=(0,\infty)$ that satisfy
\eqref{3TC} is denoted $\pf_s(0,\infty)$, with subspaces
$\pf_s(0,\infty)_\om \subset \pf_s(0,\infty)_\infty
\subset \pf_s(0,\infty)_p$ of real analytic, respectively smooth,
respectively $p$ times differentiable periodlike functions.


The main theorem in \cite{LZ} shows that $\Mf_s^0(\Gmod)$ is
isomorphic to the space of \emph{period functions}
\indind{$\pf_s(\dv)^0_\dv$; period function}{gc0}$\pf_s(\CC')^0_\om$,
characterized inside $\pf_s(\CC')_\om$ by the growth conditions
\be \label{gc0}
\ps(x) \= \begin{cases}
\oh(1)&(x\downarrow0)\,,\\
\oh(x^{-2 s})&(x\rightarrow\infty)\,.
\end{cases}\ee
These growth conditions also define $\pf_s(0,\infty)_\infty^0$ inside
$\pf_s(0,\infty)_\infty$, and $\pf_s(0,\infty)_p^0$ inside
$\pf_s(0,\infty)_p$.
(These last notations and the next are not used in~\cite{LZ}.)
In Section~3, Chap.~IV of \cite{LZ} a discussion of eigenfunctions of
the transfer operator leads to the less strict condition that there
are $c_0,c_\infty\in \CC$ such that
\be \label{gc+e}
\ps(x) \= \begin{cases}
c_0 \, x^{-1} + \oh(1)&(x\downarrow0)\,,\\
c_\infty \, x^{1-2s}+ \oh(x^{-2 s})&(x\rightarrow\infty)\,.
\end{cases}
\ee
We use this condition to define
\indlbl{gc+}{$\pf_s(0,\infty)^1_\om$}$\pf_s(0,\infty)^1_\om
\subset \pf_s(0,\infty)_\om$. In Section~3, Chapter~III of~\cite{LZ}
we see that general elements of $\pf_s(\RR_+)_\infty$ have in their
asymptotic behavior at $\infty$ an additional term $Q_\infty(x)$, and
at $0$ a term $x^{-2s}  \,  Q_0(1/x)$, with periodic functions
$Q_\infty$ and~$Q_0$. So being in $\pf_s(\RR_+)_\om^1$ is a strong
condition, almost as strong as being in $\pf_s(\RR_+)_\om^0$.
\medskip


Periodlike functions are related to cocycles. Suppose that
$\ps\in \pf_s(\RR_+)_\om$. Define $\ps$ on $(-\infty,0)$ by
$\ps(x) = - |x|^{-2s} \,  \ps(-1/x)$. One verifies that
\be \ps(x)
\= \ps(x+1) + |x+1|^{-2s} \, \ps\Bigl( \frac x{x+1} \Bigr)\ee
for all $x\in \RR\sm\{-1,0\}$ by separately considering the cases
$-1<x<0$ and $x<-1$. This extended function~$\ps$ satisfies in
$\V\fs s$
\be \ps|_{2s}\,S \= - \ps_S\,,\qquad \ps\=\ps_{2s}|\,(T+T')\,, \ee
with $T'=\matc1011$.\indlbl{Ta}{$T'=\matc1011$} These relations are
equivalent to the parabolic cocycle relations
\be \ps|_{2s}\,(1+S)\=0\,,\qquad \ps|_{2s}\,(1+ST+STST)\=0 \,. \ee
Hence $\ps \in \pf_s(\RR_+)_\om$ determines a parabolic cocycle
$c\in\Zpar^1(\Gmod;\V\fs s)$ given on the generators $S$ and~$T$ of
$\Gmod$ by $c_T=0$ and $c_S=\ps$. Conversely, a parabolic cocycle
$c \in \Zpar^1(\Gmod;\V\fs s)$ such that $c_T=0$ and such that the
set of singularities $\bsing(c_S) $ is contained in $ \{0,\infty\}$
determines a periodlike function in $\pf_s(\RR_+)_\om$ by restriction
to~$(0,\infty)$.



\begin{prop}\label{prop-Hpf}The cohomology group
$\Hpar^1(\Gmod;\V\wdgcp s) $ is isomorphic to the space
$\pf_s(\CC')_\om/\bigl
\{h|_{2s}\,(1-S)\;:\; h\in \hol(\CC)^T\bigr\}$.
\end{prop}
\begin{proof}For a given $\ps \in\pf_s(\CC')_\om$ we define
$c\in\Zpar^1(\Gmod;\V\wdgcp s)$ by
\be\label{psd} c_T\=0\,,\qquad c_S(\tau) \=
\begin{cases}
\ps(\tau) &\text{ if } \re \tau>0\,,\\
-(-\tau)^{-2s} \, \ps(-1/\tau)&\text{ if }\re \tau<0\,.
\end{cases}
\ee
If $\ps=h|_{2s}\,(1-S)$ with $h\in \hol(\CC)^T$, then
$c_S=h|_{2s}\,(1-S)$. If $c=df$, $f\in \V\wdg s$, then
$c_S=f\,|\,(S-\nobreak 1)$, and from $c_T=f\,|\,(T-\nobreak1)=0$ we
conclude that $f\,|\,T=f$, first on an excised neighborhood
of~$\RR=\proj\RR\sm\{\infty\}$ and then on~$\CC$. Thus, we obtain a
map from $\pf_s(\CC')_\om$ to $\Hpar^1(\Gmod;\V\wdgcp s) $ with
kernel $\bigl\{h\,|\,(1-\nobreak S)\;:\; h\in \hol(\CC)^T\bigr\}$.

Conversely, a given cohomology class in $\Hpar^1(\Gmod;\V\wdgcp s)$
can be represented by a cocycle such that $c_T=0$. In view of
Theorem~\ref{thmC} there is $\eta \in \V\wdgcp s$ such that the
equivalent cocycle $c-d\eta$ has values in $\V\om s$. {}From
$\eta\,|\,(1-T)\in \V\om s$ it follows that
$\bsing(\eta) \subset\{\infty\}$ and
$\bsing( c_S) \subset\{0,\infty\}$.

Restriction of $c_S$ to $(0,\infty)$ gives $\ps\in \pf_s(\RR_+)_\om$.
Moreover, $\ps$ has a holomorphic extension to a right half-plane.
The second step in the bootstrap procedure in \S4, Chap.~III of
\cite{LZ} can be applied, to see that $\ps$ extends to $\CC'$ and
provides an element of $\pf_s(\CC')_\om$.
\end{proof}

\begin{prop}
\beqabc
\begin{alignat}2 \label{Hpa}
\Hpar^1(\Gmod;\V\sa s) &\;\cong\; \pf_s(\RR_+)^0_\om\,,\\
\label{Hpb}
\Hpar^1(\Gmod;\V\smp s) &\;\cong\;
\pf_s(\RR_+)^1_\om&\quad&(s\neq\frac12)\,,\\
\label{Hpc}
\Hpar^1(\Gmod;\V\infty s) &\;\cong\; \pf_s(\RR_+)^0_\infty\,,\\
\label{Hpd}
\Hpar^1(\Gmod;\V p s) &\;\cong\; \pf_s(\RR_+)^0_p&\quad&(p\in \NN\,,\;
p\geq 2)\,.
\end{alignat}
\eeqabc
\end{prop}
\begin{proof}Each cohomology class in one of these four cohomology
groups contains a unique cocycle such that $c_T=0$. The uniqueness
follows from Propositions \ref{prop-Tinv0} and \ref{prop-Tinv1}. In
cases \eqref{Hpa} and \eqref{Hpb} we conclude that
$\bsing (c_S) \subset \{0,\infty\}$ in the same way as in the proof
of Proposition~\ref{prop-Hpf}. Restriction of $c_S$ to $(0,\infty)$
gives us $\ps\in\pf_s(\RR_+)_\om$ in cases \eqref{Hpa}
and~\eqref{Hpb}, $\ps\in \pf_s(\RR_+)_\infty$ in case~\eqref{Hpc},
and $\ps\in \pf_s(\RR_+)_p$ in case~\eqref{Hpd}. In cases \eqref{Hpa}
and \eqref{Hpc}, the fact that $c_S \in \V\infty s$ implies that
$\ps$ satisfies \eqref{gc0}. In case \eqref{Hpb}, we get \eqref{gc+e}
from Definition~\ref{V(1)def}.


Conversely, starting from a periodlike function on $(0,\infty)$, we
construct a corresponding cocycle in each of the four cases. The
hardest point is the behavior at~$0$ and~$\infty$. We have the
estimate \eqref{gc0} or~\eqref{gc+e}, and want to derive the
appropriate asymptotic behavior. We use averaging operators similar
to the transfer operator discussed in \S3 of Chap~IV in~\cite{LZ}.
{}From the three term relation
$c_S |_{2s}\,(1-\nobreak T) = c_S|_{2s}\,T'$ on $\RR \sm \{-1,0\}$ we
conclude that $c_S = c_S|_{2s}\,T'|_{2s}\,\av T^+$ on $(0,\infty)$
and $c_S=c_S|_{2s}\,T'|_{2s}\,\av T^-$ on $(-\infty,-1)$. The
asymptotic formula~\eqref{asympt} implies that $c_S$ has the desired
behavior near $\infty$ in each of the three cases, and also near~$0$
since $c_S|_{2s}\,S = - c_S$.
\end{proof}



\subsubsection{Reconstruction} A periodlike function
$\ps\in \pf_s(\CC')_\om$ determines a cohomology class in
$\Hpar^1(\Gmod;\V\wdgcp s)$
(Proposition~\ref{prop-Hpf}), which in turn determines an invariant
eigenfunction $u\in \E_s^{\Gmod}$, provided $s\neq\frac12$
(Theorem~\ref{thmC}). We want to construct $u$ directly from the
periodlike function~$\ps$.

For a period function $\ps \in \pf_s(\CC')_\om^0$ we need not use
cohomology. Proposition~2 in \S2, Chap.~I of~\cite{LZ} shows how to
associate to $\ps$ a function $f\in \hol\left(\CC\sm\RR\right)^T$
which in turn determines $\ps$. The Fourier expansion of $f$ gives
the Fourier coefficients $B_n(u)$ of~$u$, hence determines $u$
explicitly. This also works if $\ps \in \pf_s(\CC)_\om^1$. For
general periodlike functions $\ps$ there still is a holomorphic
$1$-periodic function~$f$, and its Fourier coefficients still give
the $B_n(u)$, but the coefficients $A_n(u)$ cannot be read off from
it directly. (They are hidden in the behavior of $f$ near points
of~$\QQ$.) In this case we will use the theory developed in these
notes instead.

To a given $\ps\in \pf_s(\CC')_\om$ we have associated in the proof of
Proposition~\ref{prop-Hpf} an explicit cocycle
$c\in \Zpar^1(\Gmod;\V\wdgcp s)$ by \eqref{psd}. To apply the method
in \S\ref{sect-MFsac} directly, we have to find
$h \in  \V{\om,\, \wdgwdg} s[\infty]$ such that \[c-dh\in
\Zpar^1(\Gmod;\V\om s,\V\wdgcp s) \subset Z^1(\Gmod;\V\om s)\,.\]
The existence of such a function~$h$ follows from \eqref{pcm-c}. The
proof of Proposition~\ref{prop-sing-dec} shows that the construction
of a suitable $h$ is not explicit. It seems better to apply the
method in \S\ref{sect-MFsac} directly to the cocycle
$\hat\ps \in \Hpar^1(\Gmod;\W\wdgcp s)$ given by
$\hat\ps_\gm = \tpoiss s c_\gm$. So $\hat\ps_T=0$ and
$\hat\ps_S = \tpoiss s c_S\in \W\wdgcp s \cap \W\om s[0,\infty]$.

The first step is to determine a cocycle corresponding to $\hat\ps$ in
a model of cohomology based on a tesselation. We use the fundamental
domain
\[ \fd_1 \= \bigl\{ z\in \uhp\;:\; 0\leq \re z\leq 1\,, \; |z|\geq
1\,,\; |z-1|\geq 1 \bigr\}\,\]
in Figure~\ref{figfdalt}, which differs from the standard fundamental
domain in Figure~\ref{figmodtess} on p.~\pageref{figmodtess}.
\begin{figure}[ht]
\setlength{\unitlength}{1.5cm} {\hspace*{\fill}
\begin{picture}(2,3)(-1,-.2)
\put(0,1){\circle*{.07}}
\put(1,1){\circle*{.07}}
\put(.5,.866){\circle*{.07}}
\put(-.15,1){$i$}
\put(1.1,1){$i+1$}
\put(-.6,.75){$ST^{-1}e_2$}
\put(.7,1.08){$e_2$}
\put(1.05,1.5){$e_1$}
\put(-.65,1.5){$T^{-1}e_1$}
\put(.4,.6){$\frac{1+i\sqrt 3}2$}
\put(.43,2){$\fd_1$}
\thicklines
\put(0,1){\line(0,1){1.7}}
\put(1,1){\line(0,1){1.7}}
\qbezier(.5,.866)(.8,1)(1,1)
\qbezier(.5,.866)(.3,1)(0,1)
\put(1,1.8){\vector(0,1){.3}}
\put(0,1.8){\vector(0,1){.3}}
\put(.4,.93){\vector(-4,1){.25}}
\put(.65,.93){\vector(4,1){.25}}
\end{picture}
\hspace{\fill}
\hspace{\fill}
\begin{picture}(3,3)(-.6,-.2)
\put(-1.2,0){\line(1,0){3.4}}
\put(0,1){\circle*{.07}}
\put(1,1){\circle*{.07}}
\put(.5,.866){\circle*{.07}}
\put(.43,2){$\fd_1$}
\put(-1.1,-.2){$-1$}
\put(-.03,-.2){$0$}
\put(.97,-.2){$1$}
\put(1.97,-.2){$2$}
\put(1.75,2){$C$}
\put(0,1){\line(0,1){1.7}}
\put(1,1){\line(0,1){1.7}}
\qbezier(.5,.866)(.8,1)(1,1)
\qbezier(.5,.866)(.3,1)(0,1)
\thicklines
\put(2,1){\vector(0,1){.5}}
\qbezier(-1,0)(-1,.5)(-.5,.5)
\qbezier(0,0)(0,.5)(-.5,.5)
\qbezier(0,0)(0,.5)(.5,.5)
\qbezier(1,0)(1,.5)(.5,.5)
\qbezier(1,0)(1,.5)(1.5,.5)
\qbezier(2,0)(2,.5)(1.5,.5)
\put(-1,0){\line(0,1){2.7}}
\put(2,0){\line(0,1){2.7}}
\end{picture}\hspace*{\fill}
} \caption{Fundamental domain $\fd_1$ for the modular group, and a
$1$-cycle around it.} \label{figfdalt}
\end{figure}
Here we work with parabolic cohomology, not with mixed parabolic
cohomology. So there is no need to give the neighborhood of cusps a
special treatment. In particular, we do not need an edge $f_\infty$.
We use the tesselation $\tess$ obtained from all $\Gmod$-translates
of~$\fd_1$. In Figure~\ref{figfdalt} we have indicated a
$\QQ[\Gmod]$-basis $\{e_1,e_2\}$ of $\QQ[X_1^\tess]$. The tesselation
$\tess$ is not exactly a tesselation of type \Fcoh\ as discussed
in~\S\ref{ssect-res-c}.

To find a cocycle $c \in Z^1(\Fpt^\tess;\W\wdgcp s)$ corresponding to
$\hat\ps$ we write $c=df$, with the following
$f\in \map(X_0^\tess;\W\wdg s)$. We put, with
$\rho=\frac{1+i\sqrt 3}2$,
\be f(\infty)\=0\,,\quad f(\rho)\= \frac13 \,
\hat\ps_S\,|\,(1-ST^{-1})\,,\quad f(i)\=\frac12 \, \hat\ps_S\,, \ee
and check that $f$ satisfies $f(P)\,|\,(1-\nobreak\dt)=\hat\ps_\dt$ if
$\dt \, P=P$ for $P\in \{\infty,\rho,i\}$, $\dt\in \Gm$. Next we
extend $f$ to $X_1^\tess$ by
$f(\gm^{-1}P)=f(P)\,|\,\gm + \hat\ps_\gm$ for all $\gm\in \Gm$.
Taking $\infty$ as the base point we see that $c=df$ corresponds to
the cohomology class $[\hat \ps]$.

On the $\QQ[\Gmod]$-basis $\{e_1,e_2\}$ of $F^\tess_1$:
\be c(e_1) \= \frac12 \, \hat\ps_S\,|\,T^{-1}\,,\quad c(e_2) \=
-\frac12 \, \hat \ps_S \,|\,T^{-1}+\frac13 \,
\hat\ps_S\,|\,(1-ST^{-1})\,. \ee

To represent the corresponding invariant eigenfunction $u$ on $\fd$,
we take the following $1$-cycle $C$ around $\fd$
\be C\= e_{2,\infty} + e_{\infty,-1}
+ e_{-1,0} + e_{0,1} + e_{1,2}\,, \ee
which is indicated in Figure~\ref{figfdalt}. It turns out that
\begin{align*}
 e_{0,\infty}&\= (1-S)T^{-1} e_1\,,\\
 C &\= (T^2-T^{-1}-STS+ST^{-1}S+T^2ST)
 e_{0,\infty} \\
&\=
(T^2-T^{-1}-STS+ST^{-1}S+T^2ST)(1-S)T^{-1} e_1\,.
\end{align*}
Application of the transverse Poisson transformation to the function
in \eqref{psd} gives a representative $\hat\ps_S\in \G^\wdg_s$ of
$\hat\ps_S \in \W\wdg s$ with
$\sing(\hat\ps_S) \subset i\,(0,\infty)$. We have
$c(e_{0,\infty}) = c(e_1)\,|\,T(1-\nobreak S) = \hat\ps_S$, and hence
$\sing\bigl( c(e_{0,\infty}) )
 \subset i\, (0,\infty)$. This implies
that the singularities of $\ps(C)$ are contained in the support
of~$C$. Thus, for $z\in \fd_1$ we have
\be
\begin{aligned}
u(z) &\= \frac1{\pi i} \ps(C)
\= \frac1{\pi i} \ps(e_{0,\infty}) \,|\, \bigl(
T^{-2}-T-ST^{-1}S+STS+T^{-1}ST^{-2} \bigr)\\
&\= \frac1{\pi i} \left( \hat\ps_S(z-2) -
\hat\ps_S(z+1)-\hat\ps_S\biggl(\frac z{z+1}\biggr) +
\hat\ps_S\biggl(\frac z{1-z} \biggr) + \hat\ps_S\biggl(
\frac{1-z}{z-2} \biggr) \right)\,.
\end{aligned}
\ee
Each of these values of $\hat \ps_S(z_1)$ can be expressed by a
transverse Poisson integral from $\bar z_1$ to $z_1$ of the original
periodlike function $\ps$. See~\eqref{Wdef}.


\subsubsection{The image of the invariant eigenfunctions in the first
cohomology group} In the previous section we have shown that
$\bold r \E_s^{\Gm} = \Hpar^1(\Gm;\V\om s,\V\wdg s)$ for
$s\neq\frac12$. We now show that for $\Gm=\Gm_1$ the image is not the
total first cohomology group with analytic coefficients:
\begin{prop}\label{prop-notall}Let $s\neq\frac12$. The inclusion
$\Hpar^1(\Gmod;\V\om s,\V\wdg s) \subset H^1(\Gmod;\V\om s)$ is not
an equality.
\end{prop}
\begin{proof}
We determine a cohomology class $[\ps]\in H^1(\Gmod;\V\om s)$ by
taking in the line model
\be \ps_{TS}\=0\,,\qquad\ps_T\=a\in \V\om s\quad\text{ with }a(\tau)\=
\frac{-4i \, \tau}{(1+\tau^2)^{s+1}}\,. \ee
(See the introduction of \S\ref{ssectMaf-pc} for the relations.)
The sum
\be \av T(\ps_T)(\tau) \= \sum_{n\in \ZZ} \frac{-4i \,
(\tau+n)}{\bigl(1+(\tau+n)^2\bigr)^{s+1}} \ee
converges without regularization, and has singularities in the points
of $\pm i + \ZZ$. So $\av T(\ps_T)$ does not extend as a holomorphic
$1$-periodic function on~$\CC$, which it should according to
Proposition~\ref{prop-bt-par} if $[\ps]$ were in
$\Hpar^1(\Gmod;\V\om s,\V\wdg s)= \bold r\, \E_s^{\Gmod}$.
\end{proof}



\subsubsection{Quantum Maass forms}  The notion of \emph{quantum
modular forms} was introduced by the third author based on several
examples ranging from Dedekind sums to quantum invariants of knots
\cite{Za10}. Roughly, these forms are functions $f:\QQ\to\CC$ whose
``failure of modularity'' $f - f\circ\gamma$
(or more generally $f - f|\gamma$ for some group action
$f\mapsto f|\gamma$ involving a non-trivial automorphy factor) has
some kind of continuity or analyticity property that is not shared by
the function~$f$ itself. Example~1 in~\cite{Za10} was related to the
period function, in the sense of~\cite{LZ}, of a particular Maass
form (on $\Gamma_0(2)$ and with eigenvalue~1/4), and this was
extended by the first author in~\cite{Br} with the application in
this subsection in mind.

More precisely, we will discuss how to fill the gap indicated by
a question mark in the following diagram of inclusions and
isomorphisms (in which the underlying group is the full modular
group~$\Gamma_1$, omitted from the notations, and $s\ne\frac12$) by
putting an appropriate space of quantum Maass forms in that position.
\be \label{?diag}
\begin{aligned}\text{\small $
\xymatrix@C=.55cm{ \Mf^0_s \ar@{^{(}->}@<-.4ex>[r] \ar@{<->}[d]^\cong
& \Mf^1_s\ar@{^{(}->}@<-.4ex>[r]^\neq \ar@{<->}[d]^\cong &
\E^{\Gmod}_s \ar@{<->}[d]^\cong \ar@{.}@<-.4ex>[r]& ?\ar@{.}[d]\\
\Hpar^1(\V\om s,\V\sa s) \ar@{^{(}->}@<-.4ex>[r]
& \Hpar^1(\V\om s,\V\smp s) \ar@{^{(}->}@<-.4ex>[r]^\neq
& \Hpar^1(\V\om s,\V\wdg s)\ar@{^{(}->}@<-.4ex>[r]^(.63)\neq
& H^1(\V\om s)
}$}\end{aligned} \ee




We first discuss the quantum Maass form associated to a cusp form
$u\in \Mf^0_s(\Gmod)$. In the main theorem in \cite{LZ} we associate
to $u$ among other objects a periodic holomorphic function $f_u$ on
$\CC\sm\RR$ given by
\be f_u(\tau) \= \begin{cases}
\sum_{n>0}n^{s-\frac12} A_n(u) \, e^{2\pi in\tau}&\text{ if
}\im\tau>0\,,\\
- \sum_{n<0} |n|^{s-\frac12} A_n(u) e^{2\pi i n \tau}&\text{ if
}\im\tau<0\,,
\end{cases}
\ee
with the Fourier coefficients $A_n(u)$ from the
expansion~\eqref{Fourexp-par}. In \cite{LZ}, Chap.~I, \S4, the
function $f_u$ is expressed in terms of the $L$-functions of~$u$ by
inverse Mellin transformation. By moving the line of integration in
this representation we obtain an asymptotic expansion for $f_u(iy)$
as $y\rightarrow0$. This expansion is the same for $y\downarrow0$ and
$y\uparrow0$. So $f_u$ has a smooth continuation through~$0$. Instead
of approaching $0$ vertically, we can let $\tau$ tend to $0$ along a
geodesic half-line in $\uhp$ or in~$\uhp^-$. One may show that there
exists $f_u(0)\in \CC$ such that
\be f_u(\tau) \= f_u(0) + \o(1)\quad\text{ as }\tau \ga 0\,,\ee
where $\ga$ (geodesic approach) indicates uniformity on sectors in the
upper or lower half plane bounded by geodesic half-lines.

The periodicity of $f_u$ and the formula
$f_u(\tau) - \tau^{-2s} f_u(-1/\tau) = \ps_u(\tau)$, where $\ps_u$ is
the period function associated to~$u$, leads to a unique extension of
$f_u$ to~$\QQ$ that satisfies
\begin{alignat}3 f_u(\tau) &\= f_u(\x) + \o(1)& \quad&\text{as
}\tau\ga \x&\quad&\text{for each }\x\in \QQ\,,\\
f_u|_{2s}\, \gm(\x) &\= f_u(\x) - c_\gm(\x)&&\text{for almost all
}\x\in \QQ
&&\text{for all }\gm\in \Gmod\,,
\end{alignat}
where $\gm\mapsto c_\gm$ is the $\V\sa s$-valued group cocycle
determined by $c_S = \ps_u$ on $(0,\infty)$.

The isomorphism \eqref{pcm-a} implies that there exists
$\eta\in \V\sa s$ such that
$\tilde c_\gm=c_\gm + \eta|_{2s}\,(\gm-1)$, $\gm\in \Gmod$, is a
$\V\om s$-valued cocycle on~$\Gmod$. Replacing $f_u$ by
$\tilde f_u = f_u+\eta$, we obtain the relation
$\tilde f_u|_{2s}\,\gm = \tilde f_u - \ps_\gm$ on~$\QQ$, with
$\ps \in \Zpar^1(\Gm;\V\om s,\V\sa s)$. If we add to $\tilde f_u$ an
element of $\V\om s$, then nothing essential changes. We say that
$\tilde f_u$ represents the
\indlbl{qmf}{quantum Maass form}\emph{quantum Maass form} associated
to~$u\in \Mf^0_s(\Gmod)$.
\medskip

It is argued in \cite{Br} that to define quantum Maass forms for other
invariant eigenfunctions, we should work not with functions
$\QQ \lra\CC$, but with systems of expansions, giving for each
$\x\in \QQ$ a short asymptotic expansion
\be \label{syst}
p(\x,\tau) \= \frac{d_\x}{\tau-\x} + c_\x + \o(1)\qquad (\tau\ga
\infty)\,. \ee
In the case of $u\in \Mf_s^0(\Gmod)$, the function $\tilde f_u$ gives
an system where $d_\x=0$ for all~$\x\in \QQ$. Each $\ph \in \V\om s$
(line model) defines an uninteresting example with $d_\x=0$ and
$c_\x=\ph(\x)$ for all cusps~$\x$.

The group $\Gmod$ acts on the space
\indlbl{Rsdef}{$\R_s$}$\R_s$ of all expansions as in \eqref{syst} by
\be p|_{2s}\,\matc abcd (\x,\tau) \= \bigl(
(c\tau+d)^2\bigr)^{-s} p(\gm\x,\gm\tau) + \o(1) \qquad(\tau\ga \x)\,,
\ee
with $\gm=\matc abcd\in \Gmod$. We define the $\Gmod$-module
\indlbl{Qsdef}{$\Q_s$}$\Q_s$ as the quotient in the exact sequence
\be 0 \lra \V\om s \lra \R_s \lra \Q_s\lra 0
\ee
and we call the elements of $\Q_s^{\Gmod} /\R_s^{\Gmod}$ \emph{quantum
Maass forms}, with the notation\indind{$\qMf_s(\Gmod)$}{qmfnot}
\be\label{qmfnot}
\qMf_s(\Gmod)
= \Q_s^{\Gmod} \bigm/ \R_s^{\Gmod}\,.\ee
In this way, we ignore elements of $\V\om s$ and systems in $\R_s$
that are exactly $\Gm_1$-invariant. One can show that there is an
injection $\qMf_s(\Gmod) \lra H^1(\Gm;\V\om s)$. Theorems~2 and~4 in
\cite{Br} give for $s\neq \frac12$ the following commuting diagram:
\be
\begin{aligned}&\xymatrix{ \E_s^{\Gmod} \ar@{^{(}->}[d]
\ar@{^{(}->}[dr]^{\bold r}
 \\
\qMf_s(\Gmod) \ar[r]^\cong
& H^1(\Gmod;\V\om s)
}
\end{aligned} \ee
Thus, $\qMf_s(\Gmod)$ can take the place of the question mark in the
diagram~\eqref{?diag}. Proposition~3 in~\cite{Br} relates the
vanishing of $A_0(u)$, the coefficient of $y^{1-s}$ in the Fourier
expansion of $u\in \E_s^{\Gmod}$, to the vanishing of the $d_\x$
in~\eqref{syst}.

We expect that these results can be extended to all cofinite discrete
$\Gm$ with cusps.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Maass forms and holomorphic functions}\label{sect-Mfhol}
In \S\ref{ssect-holf} we have associated to $u\in \E_s^T$ the
holomorphic function $\bt(u)\in \hol(\CC)^T$ given by
\[ \bt(u)(\z) \= \sum_{n\in \ZZ} B_n(u) \, e^{2\pi in\z}\,,\]
based on the coefficients $B_n(u)$ in the Fourier
expansion~\eqref{Fourexp-par}. To have a well defined coefficient
$B_0(u)$, we assume $s\neq\frac12$. For a $\Gm$-invariant function
$u\in \E_s^\Gm$, where $\Gm$ is a group with cusps,
$\bt(u\,|\,g_\kp)\in \E_s^T$ for each cusp $\kp\in \cu$. In the
introduction of \S\ref{sect-Mf} we have chosen the $g_\kp$ such that
$\bt(u\,|\,g_\kp)$ depends only on the class of $\kp$ in
$\Gm\backslash \cu$. Thus we are led to define\indind{$\bold j$}{jj}
\be\label{jj} \bold j:\E_s^\Gm\lra \bigoplus_{\kp\in \Gm\backslash \cu}
\hol(\CC)^T\,,\quad \bold j u = \bigl( \bt(u\,|\,g_\kp) \bigr)_\kp\,.
\ee

The kernel of $\bold j$ is, by definition, the space $\Mf^1_s(\Gm)$
introduced in \S\ref{ssect-tM}. To get information on the cokernel,
we start with $u,v\in \E_s^\Gm$ and integrate the Green's form
$\{u,v\}$ in \eqref{[]def} over the boundary of a truncated
fundamental domain $\fd_\hy$. Since $\{u,v\}$ is closed, this
integral vanishes. All edges of $\fd_\hy$, except for the edges
$f_\kp$ near the cusps $\kp\in \fdcu$, occur in $\Gm$-equivalent pairs
for which the integrals of $\{u,v\}$ cancel. Hence
\[ \sum_{\kp\in \fdcu} \int_{f_\kp} \{u,v\}\=0\,.\]
Inserting the Fourier expansions and working with Wronskians, we
obtain the following relation, valid for all $u,v\in \E_s^\Gm$:
\be\label{MSrel}
\begin{aligned}
0&\=\sum_{\kp\in \fdcu} \biggl(
 (2s-1) \, \Bigl(A_0(u\,|\,g_\kp) \, B_0(v\,|\,g_\kp) - B_0(u\,|\,g_\kp)
 \, A_0(v\,|\,g_\kp)\Bigr)\\
&\quad\hbox{}
+ \pi^{\frac12-s} \, \Gf\Bigl(s+\frac12\Bigr)
\sum_{n\neq 0} |n|^{\frac12-s} \, \Bigl(A_n(u\,|\,g_\kp) \,
B_{-n}(v\,|\,g_\kp)
- B_n(u\,|\,g_\kp)
\, A_{-n}(v\,|\,g_\kp)\Bigr)
\biggr)\,.
\end{aligned}
\ee
This is the so-called
\indlbl{MS}{Maass-Selberg relation}\emph{Maass-Selberg relation}. See,
e.g., \S3 in Chap.~IV of~\cite{Ma}.

In particular, if $v\in \Mf_s^1(\Gm)$, then
\begin{align*} 0&\=\sum_{\kp\in \fdcu} \biggl(
 (2s-1) \, B_0(u\,|\,g_\kp) \, A_0(v\,|\,g_\kp)\\
 &\qquad\hbox{}
+ \pi^{\frac12-s} \, \Gf\Bigl(s+\frac12\Bigr)
\, \sum_{n\neq 0} |n|^{\frac12-s} \, B_n(u\,|\,g_\kp) \,
A_{-n}(v\,|\,g_\kp)
\biggr)\,. \end{align*}
The right hand side of this expression makes sense if we replace the
$B_m(u\,|\,g_\kp)$ by the coefficients $b_m^\kp$ of an arbitrary
element
\[ (b_\kp)_\kp \;\in\; \bigoplus_{\kp\in \Gm\backslash\cu}
\hol(\CC)^T\,,\quad b_\kp(\z) \= \sum_{m\in \ZZ} b_m^\kp \, e^{2\pi i
m\z}\,.\]
Note that the convergence of these series implies that
\be\label{b-e} b_n^\kp \;\ll_A\; e^{-A|n|}
\quad\text{ for }n\in \ZZ\,,\; \kp\in \Gm\backslash\cu\,, \text{ for
each }A>0\,.
\ee
Thus, we have a linear map
\indlbl{mm}{$\bold m$}$\bold m $ from
$\bigoplus_{\kp\in \Gm\backslash\cu} \hol(\CC)^T$ to the dual space
$\Mf^1_s(\Gm)^\vee$.
\begin{thm}\label{thm-hex}Let $0<\re s<1$, $s\neq\frac12$.
\begin{enumerate}
\item[i)] The following sequence is exact:
\be\label{hm-seq} 0\lra \Mf^1_s(\Gm) \lra \E_s^\Gm \stackrel{\bold
j}\lra \bigoplus_{\kp\in \Gm\backslash\cu} \hol(\CC)^T \stackrel{\bold
m}\lra \Mf^1_s(\Gm)^\vee \lra 0\,. \ee
\item[ii)] Every $u\in \E_s^\Gm$ is the specialization of a family
$(u_{s'})_{s'\in U}$ of elements $u_{s'} \in \E_{s'}^\Gm$ depending
holomorphically on a parameter $s'$ on a neighborhood $U$ of~$s$.
\end{enumerate}
\end{thm}
\par\noindent\emph{Remark 1. }Part~ii) of the theorem will be used in
Chapter~\ref{ch-VI}, when we will study the relation between
$\Gm$-invariant eigenfunctions and distribution-valued cohomology.
\par\noindent\emph{Remark 2. }The proof will show that the restriction
on the spectral parameter~$s$ is non-essential.
\par\noindent\emph{Remark 3. }This theorem is essentially known if
almost all $b^\kp_n$ vanish. Since the result is peripheral to the
main themes of these notes, we will only sketch the proof.

\begin{proof}
The surjectivity of the map $\bold m$ is clear since $\Mf^1_s(\Gm)$
has finite dimension and already the restriction of $\bold m$ to
$\bigoplus_\kp \CC[q]$, with $q=e^{2\pi i \z}$, is surjective. The
Maass-Selberg relation shows that the image of $\bold j$ is contained
in the kernel of $\bold m$. The main point is to show that
$\im \bold j$ is equal to~$\ker \bold m$. We will sketch how this
follows from the spectral theory of automorphic forms. One may use
\cite{Iw} as a general reference.

We use Eisenstein series and Poincar\'e series. These converge
absolutely only for $\re s>1$. This forces us to consider also values
of $s$ outside the strip $0<\re s<1$.

For $\re s>1$, $\kp\in \cu$, the Eisenstein
series\indlbl{Eis}{Eisenstein series; $E_s^\kp$}
\be E_s^\kp (z) \= \sum_{\gm\in \Gm_\kp\backslash\Gm} (\im g_\kp^{-1} \gm
z)^s
\ee
converges absolutely and defines an element of $\E_s^\Gm$. It depends
holomorphically on $s$, and has a meromorphic continuation to
$s\in \CC$ as a family of elements of $\E_s^\Gm$. The singularities
in the region $\re s\geq \frac12$, $s\neq\frac12$ are of first order
and occur at $s=1$ and possibly at finitely many points in
$(\frac12,1)$. The latter singularities are absent for congruence
subgroups of~$\Gmod$. If $E^\kp_s$ has a singularity at
$s_0\in (\frac12,1)$ then $\res_{s_0} E^\kp_s \in \Mf^1_{s_0}(\Gm)$
and for all $u\in \Mf^1_{s_0}(\Gm)$:
\be\label{res-Eis} \int_{\Gm\backslash \uhp} u \, \res_{s_0}E^\kp_s\,
d\mu \= A_0^\kp(u)\,.
\ee

Suppose that the collection $(b_n^\kp)$ satisfies \eqref{b-e}. The
series
\[ F_s^\kp(z) \= \sum_{n\in \ZZ} b_n^\kp \, i_{s,2\pi n}(g_\kp^{-1}z)
\]
converges absolutely for all $s\in \CC$ and defines a holomorphic
family of elements of $\E_s^{\pi_\kp} \cap \W\om s[\kp]$. The
Poincar\'e series\indlbl{Poinc}{Poincar\'e series}
\be \poinc s(z) \= \sum_{\kp\in \Gm\backslash\cu}\sum_{\gm\in
\Gm_\kp\backslash\Gm} F^\kp_s(\gm z)
\ee
converges absolutely for $\re s>1$. It defines a holomorphic family of
elements of $\E_s^\Gm$, and
\be B_m(\poinc s\,|\,g_\kp) \= b^\kp_m\qquad(m\in \ZZ,\, \kp\in \fdcu)\,.
\ee


We consider the following families of $\Gm$-invariant functions, only
the first of which has values in $\E_s^\Gm$:
\[E_s\= \sum_{\kp\in \fdcu} b^\kp_0 \, E^\kp_s\,,\qquad \tilde {\poinc
s}(z) \= \sum_{\kp\in \fdcu}\sum_{\gm\in\Gm_\kp\backslash\Gm} \tilde
F^\kp_s(\gm z)\,, \]
where
\[\tilde F^\kp_s(z) \= \sum_{n\neq 0} b^\kp_n
(\im g_\kp^{-1}z)^s \, e^{2\pi i n \re(g_\kp^{-1}z)}\,. \]
This defines $\tilde {\poinc s}\in C^\infty(\Gm\backslash \uhp)$ for
$\re s>1$. The difference $P_s-E_s -\tilde {\poinc s}$ is given by a
series converging absolutely for $\re s>0$. Compared with $\poinc s$,
the advantage of $\tilde P_s$ is its square integrability. Its decay
at the cusps implies that for any Maass form $u\in \Mf_{s_1}(\Gm)$,
$0<\re s_1<1$, $s_1\neq\frac12$, the integral
$\int_{\Gm\backslash\uhp}\tilde {\poinc s}\, u \,d\mu$ converges. Its
value can be explicitly computed:
\be\label{P-int}
\int_{\Gm\backslash\uhp}\tilde {\poinc s} \, u \,d\mu \=
\sum_{k\in\fdcu} \sum_{n\neq 0} \frac{(\pi|n|)^{\frac12-s}}4
\,\Gf\Bigl(\frac{s+s_1-1}2\Bigr)\,\Gf\Bigl(\frac{s-s_1}2\Bigr)
\,A_{-n}(u\,|\,g_\kp)\, b^\kp_n\,. \ee
This quantity occurs in the expansion of $\tilde {\poinc s}$ in the
spectral decomposition of the Laplace operator in
$L^2(\Gm\backslash\uhp)$. This expansion converges absolutely for
$\re s>1$. On checks that the convergence is even better in the
region $0<\re s\leq 1$, except for the terms that have singularities
in this region. See, e.g., the reasoning in the proof of Satz~6.2 in
\cite{Ne}. For $0<\re s<1$, $s\neq\frac12$, these singularities have
at most first order and occur at values for which there are square
integrable elements of $\E_s^\Gm$ and at values at which an
Eisenstein series has a pole, in other words, at values of~$s$ for
which $\Mf^1_s(\Gm)\neq\{0\}$. This means that $\poinc s = E_s
+ \tilde {\poinc s} +\left({\poinc s}-E_s-\tilde {\poinc s}\right)$
has a meromorphic continuation to $\re s>0$, with singularities of at
most order one at the same points, and for such a point $s_0$
\[ \res_{s_0} \poinc s = \res_{s_0} E_s + \res_{s_0}\tilde {\poinc
s}\,.\]
By analytic continuation the equality $(\Dt-\ld_s) \poinc s=0$ goes
through where $\poinc s$ is holomorphic. (Work first in distribution
sense.)
Furthermore,
\be B_n(\poinc s\,|\,g_\kp) \= b^\kp_n\qquad(\kp\in \fdcu,\, n\in \ZZ)\,.
\ee


At points~$s$ for which $\Mf^1_s(\Gm)=\{0\}$, we thus have
$\poinc s\in \E_s^\Gm$ with prescribed Fourier coefficients
$B_m(\poinc s\,|\,g_\kp)$. This implies that the sequence
\eqref{hm-seq} is exact for these~$s$, and that all elements of
$\E_s^\Gm$ occur in holomorphic families on this region in the
parameter space.


It remains to consider $s_0\neq\frac12$, $0<\re s_0<1$ for which
$\Mf^1_{s_0}\neq\{0\}$. We first take $\re s_0\geq \frac12$. Then all
elements of $\Mf^1_{s_0}$ are square integrable, and
$\Mf^1_{s_0}(\Gm)\neq \{0\}$ can occur only for
$s_0 \in \frac12+i\RR$ or $s_0\in \left( \frac12,1\right)$. Hence the
space $\Mf^1_{s_0}(\Gm)$ is invariant under complex conjugation. For
$u\in \Mf^1_{s_0}(\Gm)$ we have from \eqref{res-Eis}
and~\eqref{P-int}:
\[ \int_{\Gm\backslash \uhp} u \, \res_{s_0}\poinc s \,d\mu \=
\sum_{k\in \fdcu}b_0^\kp \, A_0(u\,|\,g_\kp) + \sum_{\kp\in \fdcu}
\sum_{n\neq 0} \frac{(\pi|n|)^{\frac12-s_0}}2 \,
\Gf\Bigl(s_0-\frac12\Bigr)
\,b^\kp_n\, A_{-n}(u\,|\,g_\kp) \,. \]
This implies that the finite-dimensional space
$\Mf^1_{s_0}(\Gm)$ is spanned by residues of finitely many Poincar\'e
series $\poinc s$, for choices of the $b^\kp_n$ such that almost all
of them are zero. Hence elements of $\Mf^1_{s_0}(\Gm)$ occur as
values of holomorphic families
$s\mapsto (s-\nobreak s_0) \, \poinc s$.

Now suppose that the $b^\kp_n$ are chosen such that $\bold m (h) = 0$
for $h=(h_\kp)_\kp $, $h_\kp(\z) = \sum_n b^\kp_n \,  e^{2\pi in\z}$.
Then $\res _{s_0}\poinc s\in \Mf^1_{s_0}(\Gm)$ is orthogonal to all
$\bar u \in \Mf^1_{s_0}(\Gm)$. In other words,
$\res_{s_0}\poinc s=0$, and $s\mapsto \poinc s$ is holomorphic
at~$s=s_0$ and $\poinc{s_0}\in \E_{s_0}^\Gm$ satisfies
$\bold j \poinc{s_0} = h$. Thus, the sequence \eqref{hm-seq} is exact
for the value $s_0$ of the spectral parameter as well, and all
elements of $\E_{s_0}^\Gm$ occur as the value of a holomorphic family
of $\Gm$-invariant $\ld_s$-eigenfunctions of~$\Dt$. This finishes the
case $\frac12\leq \re s_0<1$.

Since $\E_s^\Gm=\E_{1-s}^\Gm$, all elements of $\E_s^\Gm$ with
$0<\re s<\frac12$, $s\neq\frac12$, occur as values of holomorphic
families as well.

We are left with the kernel of~$\bold m$ for $s_0$ with
$0<\re s_0<\frac12$. Suppose that the collection $(b_n^\kp)$ satisfies
\eqref{b-e} and that for all $u\in \Mf^1_{s_0}(\Gm)$
\be\label{ba} \sum_{\kp\in \fdcu} (2s_0-1) \, b_0^\kp \, A_0(u\,|\,g_\kp)
+ \pi^{\frac12-s} \, \Gf\Bigl(s+\frac12\Bigr)
\sum_{n\neq0} |n|^{\frac12-s} \, b^\kp_n \, A_{-n}(u\,|\,g_\kp)
\=0\,.\ee
As in the case $\re s_0\geq \frac12$, it follows that
$\res _{s_0} \poinc s$ is orthogonal to $\Mf^0_{s_0}(\Gm) $ which is
contained in $ \Mf^1_{s_0}(\Gm)$. If $\res_{s_0}P_s \neq 0$ it is
known that then it is a linear combination of residues at~$s_0$ of
Eisenstein series, and that $A_0(\res _{s_0}\poinc s\,|\,g_\mu)\neq0$
for some $\mu\in \fdcu$. We apply the Maass-Selberg relation
\eqref{MSrel} to $E^\ld_s$ and the Poincar\'e series $P^{\kp,n}_s$,
which is the Poincar\'e series with $b^\kp_n=1$ and all other
$b_{n'}^{\kp'}$ equal to zero. This gives
\[ 2 \, A_0(P^{\kp,n}_s\,|\,g_\ld) \=
(\pi|n|)^{\frac12-s}\,\Gf\Bigl(s-\frac12\Bigr)\,
A_{-n}(E^\ld_s\,|\,g_\kp)\,.\]
Application of \eqref{MSrel} to $E_s^\kp$ and $E^\mu_s$ gives
$A_0(E^\kp_s\,|\,g_\mu) \= A_0(E^\mu_s\,|\,g_\kp)$, which is the
symmetry of the scattering matrix. Hence we have
\begin{align*}
A_0(&\res_{s_0} \poinc s\,|\,g_\mu) \= \sum_\kp b_0^\kp\, \res_{s_0}
A_0(E^\kp_s\,|\,g_\mu) + \sum_\kp \sum_{n\neq 0} b_n^\kp\, \res_{s_0}
A_0(P^{\kp,n}_s\,|\,g_\mu)
\displaybreak[0]\\
&\= \sum_\kp b_0^\kp\, A_0\bigl(\res_{s_0}E^\mu_s\,|\,g_\kp\bigr) +
\sum_\kp \sum_{n\neq 0} \frac{(\pi|n|)^{\frac12-s_0}}2
\,\Gf\Bigl(s_0-\frac12\Bigr)\,
A_{-n}\bigl(\res_{s_0}E^\mu_s\,|\,g_\kp\bigr)\\
&\=0\,,
\end{align*}
by assumption \eqref{ba} on the $b_n^\kp$. Hence
$\res_{s_0}\poinc s=0$, and $\poinc{s_0}\in \E_{s_0}^\Gm$ has the
prescribed Fourier coefficients $B_n(\poinc{s_0}\,|\,g_\kp) = b^\kp_n$.
This implies that the sequence \eqref{hm-seq} is exact
for~$s=s_0$.\medskip

Part~ii) is a consequence of part~i). For instance, if $s$ does not
belong to the
(discrete)
set when $\Mf^1_s(\Gm)$ is non-zero, then we can take $u_{s'}$ to be
the preimage of any holomorphic variation of
$\bold j u \in \bigoplus_{\kp\in \Gm\backslash \cu} \hol(\CC)^T$,
e.g., the constant family $s'\mapsto \bigl( \bt(u\,|\,g_\kp)
\bigr)_\kp$.

In the proof of part~i) we have seen that for any
$\bigl( \sum_n b_n^\kp e^{2\pi i n \z} )_\kp$ in the kernel of
$\bold m$ at $s$ there is a family of Poincar\'e series
$s'\mapsto P_{s'}$ such that $\bold j P_s  $ or
$\bold j \res_{s'=s} P_{s'}$ is equal to
$ \bigl( \sum_n b_n^\kp e^{2\pi i n \z} )_\kp$.
\end{proof}






\skipbetweenchapters
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%   Chapter 5   smooth cohomology                         %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\sect{Maass forms and differentiable cohomology}\label{ch-V}



% special equation numbering in the introduction of the chapter
\setcounter{equation}{0}
\renewcommand\theequation{V.\arabic{equation}}

In the chapters \ref{ch-II}--\ref{ch-IV} we studied the relation
between invariant eigenfunctions and (semi)-analytic cohomology, and
proved Theorem~\ref{thmC} and most of the statements in Theorems
\ref{thmA} and~\ref{thmB}. In this chapter we turn to smooth and
differentiable cohomology. We will give a proof of the isomorphisms
$\Hpar^1(\Gm;\V\sa s) \cong \Hpar^1(\Gm;\V\infty s) \cong
\Hpar^1(\Gm;\V p s)$, $p\geq 3$, in Theorem~\ref{thmB}. Actually, we
will show that
\be\label{MfH1i} \Mf^0_s(\Gm) \; \cong\; \Hpar^1(\Gm;\V\infty s)\,,\ee
and
\be \label{MfH1p} \Mf^0_s(\Gm) \;\cong\; \Hpar ^1(\Gm;\V p
s)\qquad(p\in \NN\,,\; p\geq 3)\,,\ee
which together with the previously proved results (in particular,
\eqref{Pal-sa} and~\eqref{pcm-a}) proves the remaining isomorphisms
in Theorem~\ref{thmB}.


The isomorphism \eqref{MfH1p} will be established in \S\ref{sect-dpc}
by a method analogous to that used in \S\ref{ssect-coie}, with
adaptions to the differentiable context. In \S\ref{sect-smcoh} this
leads to Theorem~\ref{thm-sm}, which gives the
isomorphism~\eqref{MfH1i}. A consequence is that the space of modular
Maass cusp forms is in bijective correspondence with the space of
smooth period functions. This result, Theorem~\ref{thm-pfsm}, extends
the main result in~\cite{LZ}.

Most of the proofs in this chapter work for general cofinite discrete
subgroups of $\PSL_2(\RR)$. At the end of Section~\ref{sect-dpc} we
use separate approaches for cocompact groups and groups with cusps.
The cocompact case is the harder one. As in \S~\ref{ssect-injcc} in
Chapter~\ref{ch-II} we need for cocompact groups to extend cocycles
to hyperbolic fixed points.



The isomorphy of $\E_s^\Gm$ and $H^1(\Gm;\V\infty s)$ in the cocompact
case is already known, though with a quite different proof, from the
work of Bunke and Olbrich~\cite{BO95}.


% now use the standard equation numbering again
\renewcommand\theequation{\equationstandard}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Differentiable parabolic cohomology}\label{sect-dpc}

In this section we relate the cohomology group $\Hpar^1(\Gm;\V p s)$,
$p\in \NN$, $p\geq 2$, to the space of Maass cusp forms
$\Mf_s^0(\Gm)$. The space $\V p s$ was defined in
\S\ref{ssect-mps-bd}. Recall that in Chapters \ref{ch-II}
and~\ref{ch-IV} the map from parabolic cohomology to cusp forms was
constructed, not with cohomology with values in the space $\V\om s$
of analytic functions on $\p\H$, but in the isomorphic space
$\W\om s$ of boundary germs of $\ld_s$-eigenfunctions of the Laplace
operator~$\Dt$. Here we work with the corresponding space $\W p s$ of
boundary jets, as defined in~\S\ref{ssect-bj}.

The construction of Maass forms from cocycles in \S\ref{ssect-coMacc}
and \S\ref{ssect-coie} used locally finite sums. To generalize this,
in~\S\ref{ssect-ccd}, to differentiable cocycles we will need
infinite sums. The resulting convergence questions require some
geometric considerations, carried out in~\S\ref{ssect-gp}. The proof
for discrete groups with cusps is completed
in~\S\ref{ssect-Mcfdcocyc}, where we prove that if $p\geq 3$ the map
that we have given is injective with image in the cusp forms.
(The surjectivity will be an easy consequence of that in the analytic
case.)
The injectivity in the cocompact case is handled
in~\S\ref{ssect-dc-cc}.






\subsubsection{Construction of a Maass form from a given
cocycle}\label{ssect-ccd}
We start with a cocycle $\ps \in Z^1(\Fpt^\tess;\W p s)$ with
$p\in \NN$, $p\geq2$, and use a tesselation $\tess$ of type~\Fcoh.
(See \S\ref{ssect-cc-ex} and~\S\ref{ssect-res-c}.) We choose a
$\Gm$-equivariant lift $\tilde\ps \in C^1(\Fpt^\tess ,\G^p_s)$ of
$\ps$, corresponding to the exact sequence
  \be\label{exs-NGW} 0\lra \N^p_s \lra \G^p_s\lra \W p s\lra 0\;, \ee
with the space $\G^p_s$ of representatives and the space $\N^p_s$ of
functions with decay of order~$p$ at the boundary, as introduced
in~\S\ref{ssect-bj}. Then $d\tilde\ps\in C^2(\Fpt^\tess,\N_s^p)$ and
we put
\be\label{upsavp}
 u_\ps(z)
 \= \frac1{\pi i} \, \sum_{\gm\in \Gm} d\tilde\ps(\fd)(\gm z)
 \= \av \Gm d\tilde\ps(\fd)(z)
 \qquad(z\in \H)\,. \ee
\begin{prop}\label{prop-upsp}The series in~\eqref{upsavp} converges
absolutely and uniformly on compact sets in~$\uhp$. The sum it
defines belongs to $ \E_s^\Gm$, and does not depend on the choice of
the lift~$\tilde\ps$ or on the choice of the representative $\ps$ in
the cohomology class $[\ps] \in \Hpar^1(\Gm;\W p s)$.


If $Z\in  \ZZ[X_1^\tess]$ is a cycle consisting only of edges that are
$\Gm$-translates of edges in $\p\fd$ that describes a simple
positively oriented closed curve, then
\be\label{upsavp-C} u_\ps \= \frac1{\pi i}\, \Bigl( \tilde\ps(Z)
+ \sum_{\gm\in \Gm,\, \gm^{-1}\fd\;\text{ \rm outside}\;Z}
d\tilde\ps(\fd)|\gm\Bigr)\,.
\ee
\end{prop}
\begin{proof}The function $h=d\tilde\ps(\fd)$ is in~$\N_s^p$ and hence
satisfies $h(z) \= o\bigl( \bigl( \frac y{|z+i|^2}\bigr)^{s+p}\bigr)$
as $z\rightarrow \p\uhp$. Since
$\frac y{|z+i|^2} = \bigl(2+\nobreak2\cosh \dist(z,i)\bigr)^{-1}$
(Table~\ref{uhp-disk} in~\S\ref{ssect-hp}), we have $h(z) =
o\bigl(e^{-(s+p)\,\dist(z,i)}\bigr)$ as $z\rightarrow \p\uhp$.


Let $K\subset\H$ be a compact set. For $z\in K$ the number of
$\gm\in \Gm$ such that $\dist(\gm z,i) \leq R$ is at most
$ \oh_K(e^R) $ since the area of a hyperbolic circle with large
radius~$R$ is asymptotic to $4\pi e^R$. We get
\[ \sum_{\gm\in \Gm}\bigl|h(\gm z)\bigr| \;\ll_K\; \sum_{R=1}^\infty
e^R e^{-(s+p)R } \;<\; \infty\,.\]
This proves the first statement of the proposition.

Our next observation, which will be used repeatedly, is that we have
\[ \tilde\ps(Z) \= \sum_{\gm\in \Gm,\; \gm^{-1}\fd\textrm{ inside }Z}
h|\gm\]
for any cycle $Z$ as in the proposition, since $h=d\tilde\ps$. This
gives the expression~\eqref{upsavp-C} for~$u_\ps$. It also follows
that $u_\ps$ is the limit of $\frac1{\pi i}\, \tilde\ps(Z_R)$ for any
sequence of cycles $\{Z_R)\}_{R\in \NN}$ approaching the boundary,
for instance those given in Lemma~\ref{lem-C-dt} below, where $Z_R$
has distance at least $R$ to~$i$ and consists of $\oh(e^R)$ edges.
This observation is useful both to prove that $u_\ps$ is independent
of the choice of the lift~$\tilde\ps$ and that it is an
eigenfunction. For the former we observe that changing $\tilde\ps$ to
$\tilde\ps+\ch$ with $\ch\in C^1(\Fpt^\tess;\N_s^p)$ each edge $e$
in~$Z_R$ contributes at most $o(e^{-(s+p)R})$ to $\ch(Z_R)$, which
gives $\ch(Z_R) = \oh(e^R)\, o(e^{-(s+p)R}) $. Hence
$\lim_{R\rightarrow\infty}\ch(Z_R) = 0$.


The definition of $\W s p$ implies that
$(\Dt-\nobreak\ld_s) \tilde\ps \in C^1(\Fpt^\tess;\N_s^p)$. It
follows that with $\{Z_R\}$ as above we have $(\Dt-\nobreak \ld_s)
\, \tilde\ps(Z_R)= \oh\bigl( e^R\, e^{-(s+p)R})=o(1)$, this estimate
being uniform on compact sets. Hence we have with a test function
$\th \in C_c^\infty(\uhp)$
\begin{align*}
\langle \th,\,&(\Dt-\ld_s) u_\ps \rangle \= \int_\uhp
\bigl((\Dt-\ld_s) \th\bigr) \; u_{\ps} \, d\mu
\qquad(\text{since $\Dt-\ld_s$ is self-adjoint})
\\
&\= \frac1{\pi i}\,\lim_{R\rightarrow\infty} \int_\uhp
\bigl((\Dt-\ld_s)\th\bigr)\; \tilde\ps(Z_R)\, d\mu
\qquad(\text{since $\th$ is compactly supported})
\\
&\= \frac1{\pi i}\,\lim_{R\rightarrow\infty} \int_\uhp \th \;
   \bigl((\Dt-\ld_s)\tilde\ps(Z_R)\bigr)\, d\mu
\qquad(\text{again since $\Dt-\ld_s$ is self-adjoint})
\\
&\= \lim_{R\rightarrow\infty} \oh\biggl(
\int_\uhp|\th|\,d\mu\;\sup_{z\in \supp
\th}\bigl|(\Dt-\ld_s)\tilde\ps(Z_R)(z)\bigr|\biggr)\\
& \=0\qquad
(\text{by the estimate \eqref{Dts-s-est-a}})\,.
\end{align*}
So $(\Dt-\nobreak\ld_s) u_\ps =0$ weakly. By elliptic regularity,
$(\Dt-\ld_s) u_\ps =0$ holds also at the level of functions. This
shows that $u_\ps\in \E_s$, and the $\Gm$-invariance is obvious.

Finally, we add to $\ps$ a coboundary $df$ with
$f\in C^0(\Fpt^\tess;\W p s)$. We can lift $f$ to
$\tilde f\in C^0(\Fpt^\tess;\G_s^p)$, and change the lift~$\tilde\ps$
of $\ps$ to the lift $\tilde\ps+d\tilde f$ of $\ps_f$. This changes
$d\tilde\ps(\fd)$ by $ d d f(\fd)=0$, and does not influence the
definition of~$u_\ps$.
\end{proof}


Thus we have defined a map
\indlbl{alp}{$\al_s^p$}$\al_s^p : [\ps] \mapsto u_\ps$ from
$\Hpar^1(\Gm;\W p s) \lra \E_s^\Gm$. If $q\in\nobreak \NN$,
$q\geq p$, then $\ps \in Z^1(\Fpt^\tess;\W q s)$ also determines an
element in $Z^1(\Fpt^\tess;\W p s)$. The construction of $u_\ps$
shows that $\al_s^q[\ps]=\al_s^p[\ps]$. For
$\ps\in Z^1(\Fpt^\tess;\W\om s)$ the sequence $\tilde\ps(Z_R)(z)$
stabilizes, uniformly for $z$ in compact sets, and this shows that
$u_\ps$ coincides with $u_\ps$ defined in \S\ref{ssect-coMacc}
and~\S\ref{ssect-coie}. Thus $\al_s^p[\ps] = \al_s^\om[\ps]$ on
$\Hpar^1(\Gm;\W\om s)$, with $\al_s^\om$ as in
Theorem~\ref{thm-cc-upsi} and Proposition~\ref{prop-al-cp}. This
implies that in the differentiable case we also have
$u_{\bold q u}=u$ for $u\in \E^\Gm_s$, where
$\bold q: \E_s^\Gm\rightarrow H^1(\Gm;\W\om s)$ is the map
constructed in \S\ref{ssect-Maf-coh}, in the version given
after~\eqref{q-geom}. Summarizing, we have:
\begin{prop}\label{prop-alp}
For each $p,q\in \NN$, $q\geq p\geq 2$ there are linear maps $\al_s^p$
and $\al_s^q$ induced by $\ps\mapsto u_\ps$ such that the following
diagram commutes:
\[ \xymatrix{
& \Mf^0_s(\Gm) \ar@{^{(}->}[r] & \E^\Gm_s
\\
\Hpar^1(\W\om s,\W\sacp s)\ar[r]_{\cong} \ar[ur]_{\al_s^\om}^\cong &
\Hpar^1(\W\sacp s) \ar[r] & \Hpar(\W q s) \ar[r] \ar[u]^{\al_s^q} &
\Hpar(\W p s)
\ar[ul]^{\al_s^p} } \]
\end{prop}
We have suppressed $\Gm$ in the notation for cohomology groups. See
\eqref{pcm-a} for the isomorphism
$\Hpar^1(\Gm;\W\om s,\W\sacp s)\lra \Hpar^1(\Gm;\W\sacp s)$.

It remains to be shown that the linear map $\al_s^p$ is injective,
and, if $\Gm$ has cusps, that its image is in the space of cusp
forms. Before turning to that question, in \S\ref{ssect-Mcfdcocyc}
and \S\ref{ssect-dc-cc}, we prove the geometrical result that we used
in the proof of Proposition~\ref{prop-upsp}.

\subsubsection{Geometrical lemmas}\label{ssect-gp}
The result that we used is the following:
\begin{lem}\label{lem-C-dt}Let $R>0$. There exists a cycle
$Z=Z_R\in \ZZ[X_1^\tess]$ consisting of $\oh(e^R)$ edges contained in
$\Gm \p\fd$ and lying outside the open hyperbolic disk with center
$i$ and radius~$R$, with winding number~$1$ around~$i$.
\end{lem}
\begin{remarks}\emph{Remark. } The circumference of a hyperbolic cycle
with radius $R$ is approximately $2\pi \, e^R$ as
$R\rightarrow\infty$. The lemma says that the conditions on $Z$ do
not force it to have substantially more edges that is to be expected
from the length of a hyperbolic circle.
\end{remarks}


The proof of Lemma~\ref{lem-C-dt} will be very simple if $\Gm$ has no
cusps, but if $\Gm$ has cusps and $R$ is large, the path
corresponding to~$Z$ will always have to go through some cusps, and
in that case we will need a bound for the number of times that the
curve $Z$ is forced to go through a cusp. We first estimate this
quantity.

We recall that for groups with cusps the fundamental domain has a
decomposition $\fd= \fd_\hy \cup \bigcup_{\kp\in \fdcu} V_\kp$ where
$\fd_\hy$ is compact and where $V_\kp$ is the closure of the
intersection of $\fd$ with the open horocyclic disk $D_\hy(\kp)$
in~\eqref{horocycle}. The boundary of $D_\hy(\kp)$ is the horocycle
$H_\hy(\kp)$.

\begin{lem}\label{lem-horest}The number of horocycles
$\gm^{-1} H_\hy(\kp)$, $\gm\in \Gm$, $\kp\in \fdcu$, that intersect the
hyperbolic circle $C_R$ around~$i$ with radius $R$ is $\oh(e^R)$ as
$R\rightarrow\infty$. The number of cuspidal triangles
$\gm^{-1}V_\kp$, $\gm\in \Gm$, $\kp\in \fdcu$, that intersect the
circle $C_R$ is also $\oh(e^R)$ as $R\rightarrow\infty$.
\end{lem}
\begin{proof}
$\fdcu$ is finite, so we may restrict ourselves to considering one
cusp $\kp\in \fdcu$, which we conjugate to~$\infty$.

In the counting of the horocycles, $\gm$ runs over
$\Gm_\infty\backslash\Gm$. We write $\gm^{-1} = \matc abcd$. The
horocycle is determined by the first column
$\begin{pmatrix}a\\c\end{pmatrix}$ of $\gm^{-1}$. The maximum of
\[x\mapsto
\frac{\im\bigl(\gm^{-1}(i\hy+x)\bigr)}{|\gm^{-1}(i\hy+x)+i|^2} =
\frac \hy{(ax+b-c\hy)^2+(a\hy+cx+d)^2}\]
occurs for $x=\frac{-ab-cd}{a^2+c^2}$, and has value $\frac{(a^2+c^2)
\, \hy}{\left(1+(a^2+c^2) \, \hy\right)^2} \leq \frac1{(a^2+c^2) \, \hy}$.
Since $\frac y{|z+i|^2} \sim e^{-R}$ on~$C_R$, the number of
horocycles intersecting $C_R$ is bounded by $N\bigl(e^R/\hy\bigr)$,
where
\be\label{Nkpdef}
 N(B) \;:=\; \# \Bigl\{\,\gm\in \Gm_\infty\backslash\Gm\;:\;
 \gm\not\in \Gm_\infty\,,\;a^2+c^2 \leq B\,\Bigr\}\;.
 \ee
Lemma 2.10 in \cite{Iw}, applied with $z=i$ and $g_{\mathfrak a}=1$,
implies that $N(B)
=\oh\bigl( B \bigr)$ as $B\rightarrow\infty$. This gives the first
statement in the lemma.

For a fixed horocycle $\gm^{-1} H_\hy(\kp)$, there may be many cuspidal
triangles $\gm^{-1} V_\infty$ intersecting $C_R$. Their number
differs by no more than $2$ from the number of intervals
$\bigl[-\frac12+n,\frac12+n\bigr]$, $n\in \ZZ$, containing an element
$x$ with
$\frac{\im(\gm^{-1}(i\hy+x)}{|\gm^{-1}(i\hy+x)+i|^2} \geq e^{-R}$. If
such $x$ occur, the equation
$\left(ax+b-c\hy\right)^2+\left(a\hy+cx+d\right)^2 = \hy e^R$ has
solutions $x_1$ and $x_2$. These solutions satisfy
\[ (x_2-x_1)^2 \= \frac 4{a^2+c^2} \, \bigl( e^R-2\bigr) \, \hy -
4\hy^2-\frac 4{(a^2+c^2)^2} \leq \frac
{4\hy(1-2e^{-R})}{(a^2+c^2)e^{-R}} \;\ll\; \frac{\hy e^R}{a^2+c^2}
\,.\]
So the number of sectors is bounded by
\[ 2 + |x_2-x_1| \;\ll\; \frac{\sqrt{\hy e^R}}{\sqrt{a^2+c^2}} \,.\]
For $\gm \in \Gm_\infty$ this gives $\oh\left( \sqrt{\hy e^R}\right)$
sectors. The number of other sectors to be counted is estimated by
\begin{align*}
\sum_{\begin{array}{c}\scriptstyle\gm\in \Gm_\infty\backslash\Gm,\,
\gm\not\in \Gm_\infty,\\
\scriptstyle a^2+c^2\leq e^R/\hy \end{array}}
\frac{\sqrt{\hy e^R}}{\sqrt{a^2+c^2}}
&\;\ll\; \sum_{l=0}^\infty \sum_{
\begin{array}{c}\scriptstyle\gm\in \Gm_\infty\backslash\Gm,\,
\gm\not\in \Gm_\infty,\\
\scriptstyle2^{-l-1} e^R/\hy\leq a^2+c^2\leq 2^{-l}e^R/\hy
\end{array}} \frac{\sqrt{\hy e^R}}{2^{-(l+1)/2}\sqrt{e^R/Y}}
\\
&\;\ll\;\hy\,\sum_{l=0}^\infty 2^{(l+1)/2}\, N(2^{-l}\,e^R/\hy)
\;\ll\; e^R \,,\end{align*}
where in the last line we have again used $N(B)=\oh(B)$.
\end{proof}


\begin{proof}[Proof of Lemma~\ref{lem-C-dt}]Let
$\Gm(R) = \bigl\{\gm\in \Gm\;:\;
\gm^{-1}\fd\cap D_R\neq\emptyset\bigr\}$, where $D_R$ denotes the
hyperbolic disk around~$i$ with radius~$R$. We put
$A_R = \bigcup_{\gm\in \Gm(R)} \gm^{-1}\fd$, and take $Z = \p A_R$.
Since $\p\fd$ has finitely many edges, $Z$ consists of at most
$\oh \bigl( \# \Gm(R)\bigr)$ edges. Each edge occurring in $Z$ (with
non-zero factor)
has a distance at least $R$ to~$i$. The curve $Z$ encircles $D_R$ once
in the positive direction. To complete the proof we show that
$\# \Gm(R) = \oh(e^R)$ as $R\rightarrow\infty$.

We use that $\fd$ is the union of a compact set $\fd_\hy$ and finitely
many cuspidal triangles $V_\kp$, with $\kp\in \fdcu$.
Lemma~\ref{lem-horest} estimates the number of $\gm\in\Gm$ such that
$\gm^{-1}V_\kp$ intersects $D_R$ for some $\kp\in \fdcu$ by~$\oh(e^R)$.
To count the number of $\gm^{-1}\fd_\hy$ intersecting $D_R$ we note
that the distance between any two points of $\gm^{-1}\fd_\hy$ is
bounded by some number~$t$, independently of $\gm\in \Gm$. Hence if
$\gm^{-1}\fd_\hy$ intersects $D_R$, then $\gm^{-1}\fd_\hy$ is
contained in $D_{R+t}$. This leads to an estimate by
\[ \frac{\mathrm{area}(D_{R+t})}{\mathrm{area}(\fd_\hy)} \=
\oh(e^{R+t}) \= \oh(e^R)\,. \qedhere\]
\end{proof}

In the next subsection we will use the following modification of
Lemma~\ref{lem-C-dt}:
\begin{lem}\label{lem-C-dt-cu}Suppose that $\x$ and $\eta$ are cusps
of~$\Gm$. Let $R>0$, and denote by $D_R$ the open hyperbolic disk
around~$i$ with radius~$R$. There exists a chain
$A=A_R\in \ZZ[X_1^\tess]$ consisting of $\oh_{\x,\eta}(e^R)$ edges in
$\Gm\p\fd$ that describes a path from $\x$ to~$\eta$ in the region
$\uhp\sm D_R$ and is homotopic in $\bar\uhp\sm D_R$ to the (oriented)
arc in~$\proj\RR$ from $\x$ to~$\eta$.
\end{lem}
\begin{proof}Let $C\in \ZZ[X_1^\tess]$ denote a path from $\xi$ to
$\eta$ along edges of the tesselation. It consists of
$\oh_{\x,\eta}(1)$ edges. Let $Z_R$ be a cycle as in
Lemma~\ref{lem-C-dt}, consisting of $\oh(e^R)$ edges and encircling
$D_R$ once.
\begin{center} \setlength{\unitlength}{1cm}
\begin{picture}(5.4,5.4)(-2.7,-2.7)
% circles
\put(0,0){\circf}
%\put(0,0){\circth}
%\put(1,-.1){$D_R$}
% points
\put(-0.772542,2.37764){\circle*{.1}}
\put(-0.772542,-2.37764){\circle*{.1}}
\put(-.9,2.6){$\x$}
\put(-.9,-2.7){$\eta$}
% direct path
\qbezier(-0.772542,2.37764)(-.5,2)(-.3,1.8)
\qbezier(-0.772542,-2.37764)(-.5,-2)(-.3,-1.8)
\qbezier(-.3,1.8)(0,.1)(-.3,-1.8)
\put(-.1,-.1){$C$}
% cycle
\put(1,1.45){$Z_R$}
\qbezier(-.3,1.8)(-1.5,1)(-2,.1)
\qbezier(-2,.1)(-2,0)(-2.5,0)
\qbezier(-2,-.1)(-2,0)(-2.5,0)
\qbezier(-2,-.1)(-1.5,-1.1)(-.3,-1.8)
%
\qbezier(-.3,-1.8)(-0,-2)(.4,-1.8)
\qbezier(.4,-1.8)(1.2,-1.1)(2.1,0)
\qbezier(2.1,0)(2.1,.1)(2.47,.3)
\qbezier(2.1,.4)(2.3,.3)(2.47,.3)
\qbezier(0,2.3)(1,1)(2.1,.4)
\qbezier(0,2.3)(-.1,1.9)(-.3,1.8)
\thicklines
% arc on boundary
\qbezier(-0.772542,2.37764)(-1.54138,2.12153)(-2.02254,1.46946)
\qbezier(-2.02254,1.46946)(-2.49401,0.810353)(-2.5,0)
\qbezier(-2.5,0)(-2.49401,-0.810353)(-2.02254,-1.46946)
\qbezier(-2.02254,-1.46946)(-1.54138,-2.12153)(-0.772542,-2.37764)
\end{picture}
\qquad
\begin{picture}(5.4,5.4)(-2.7,-2.7)
% circles
\put(0,0){\circf}
\put(0,0){\circth}
\put(1,-.1){$D_R$}
% points
\put(-0.772542,2.37764){\circle*{.1}}
\put(-0.772542,-2.37764){\circle*{.1}}
\put(-.9,2.6){$\x$}
\put(-.9,-2.7){$\eta$}
\thicklines
% parts from Z_R
\qbezier(-.3,1.8)(-1.5,1)(-2,.1)
\qbezier(-2,.1)(-2,0)(-2.5,0)
\qbezier(-2,-.1)(-2,0)(-2.5,0)
\qbezier(-2,-.1)(-1.5,-1.1)(-.3,-1.8)
% parts from C
\qbezier(-0.772542,2.37764)(-.5,2)(-.3,1.8)
\qbezier(-0.772542,-2.37764)(-.5,-2)(-.3,-1.8)
% arc on boundary
\qbezier(-0.772542,2.37764)(-1.54138,2.12153)(-2.02254,1.46946)
\qbezier(-2.02254,1.46946)(-2.49401,0.810353)(-2.5,0)
\qbezier(-2.5,0)(-2.49401,-0.810353)(-2.02254,-1.46946)
\qbezier(-2.02254,-1.46946)(-1.54138,-2.12153)(-0.772542,-2.37764)
\end{picture}
\end{center}
The cycle $Z_R$ and the direct path intersect each other in points of
$X_0^\tess$. As in the sketch on the right, this leads to a path with
the desired properties, with $\oh_{\x,\eta}(1)+
\oh(e^R)+\oh_{\x,\eta}(1)$ edges, going from $\x$ along $C$ to the
first intersection point, then counterclockwise along $Z_R$ to the
last intersection point, and then along $C$ to~$\eta$.
\end{proof}








\subsubsection{Maass cusp forms associated to differentiable parabolic
cohomology classes}\label{ssect-Mcfdcocyc} In the cocompact case,
$\Mf^0_s(\Gm) = \E_s^\Gm$, and parabolic cohomology coincides with
standard cohomology. In this case it only remains to show that the
maps $\al_s^p$ are injective. If $\Gm$ has cusps, we need to show not
only that $\al_s^p$ is injective, but also that its image is in the
space of Maass cusp forms $\Mf^0_s(\Gm)$. Surprisingly, the presence
of cusps actually helps in proving the injectivity.

The cocompact case will be discussed in the next subsection. Here we
suppose that $\Gm$ has cusps. We use a tesselation $\tess$ of
type~\Fcoh. (See~\S\ref{ssect-res-c}.)

Let $u=u_\ps\in \E_s^\Gm$ be the invariant eigenfunction associated to
the cocycle $\ps\in Z^1(\Fpt^\tess;\W s p)$ via a lift
$\tilde\ps \in C^1(\Fpt^\tess;\G_s^p)$. We use the following variant
of the formula~\eqref{upsavp-C} in Proposition~\ref{prop-upsp}: if
$\x$ and $\eta$ are distinct cusps of~$\Gm$, we have the splitting
\be\label{ups-decomp}
u=u_{\x,\eta}+u_{\eta,\x}\ee
with
\be\label{uxieta}
\begin{aligned}
u_{\x,\eta} &\= \frac1{\pi i}\, \biggl( \tilde\ps(A)+ \sum_{\gm\in
\Gm,\, \gm^{-1}\fd\;\text{to the right of}\; A} d\tilde\ps(\fd)|\gm
\biggr)\,,\\
u_{\eta,\x} &\= \frac1{\pi i}\, \biggl(
-\tilde\ps(A)+ \sum_{\gm\in \Gm,\, \gm^{-1}\fd\;\text{to the left
of}\; A} d\tilde\ps(\fd)|\gm \biggr)\,,
\end{aligned}
\ee
where the chain $A\in Z[X_1^\tess]$ represents a path from $\x$
to~$\eta$ without self-intersections, following edges contained in
$\Gm\,\partial\fd$.
The words ``to the left'' and ``to the right'' of~$A$ depend on the
orientation of $A$: ``to the left of~$A$'' is equivalent to ``to the
right of the opposite path~$-A$.'' The definitions of $u_{\x,\eta}$
and $u_{\eta,\x}$ do not depend on the choice of path from $\x$
to~$\eta$ for the same reason that the right hand side
of~\eqref{upsavp-C} was independent of the chosen path~$A$.
\begin{prop}
For cusps $\x,\eta,\th$ satisfying $\x<\eta<\th<\x$ for the cyclic
order of~$\p\H$, and $\gm\in \Gm$ we have
\beqabc\begin{align}
\label{ucoc} u_{\x,\th}&\= u_{\x,\eta}+u_{\eta,\th} \,,\\
\label{ugm}
u_{\gm^{-1}\x,\gm^{-1}\eta} &\= u_{\x,\eta}|\gm\,, \\
\label{Uef}
u_{\x,\eta}&\,\in \,\E_s\,.
\end{align}\eeqabc
\end{prop}
\begin{proof}
The statements in \eqref{ucoc} and~\eqref{ugm} are clear from the
definitions. For \eqref{Uef} we proceed as in the proof that
$u\in \E_s$ in Proposition~\ref{prop-upsp}, now with a sequence of
paths $A_R$ from $\x$ to~$\eta$ as in Lemma~\ref{lem-C-dt-cu}.
\end{proof}


The following lemma implies that $u_{\x,\eta}$ is relatively small
near the arc from $\eta$ to~$\x$ in~$\p\uhp$.
\begin{lem}\label{lemxieta}Let $\ps$, $\tilde\ps$ and $u_{\x,\eta}$ be
as above. We denote by $g=g_{\x,\eta}$ the geodesic from $\x$
to~$\eta$. For any choice of path $A$ from $\x$ to $\eta$ we have
\be\label{xieta}
\begin{aligned}
\sum_{\gm\in \Gm,\, \gm^{-1}\fd\;\text{\rm to the right of}\; A}
d\tilde\ps(\fd)(\gm z)
&\= \oh \Bigl( \Bigl( \frac y{|z+i|^2}\Bigr)^{s+p-2}\Bigr)\\
\text{and }\quad u_{\x,\eta}(z)&\= \oh \Bigl( \Bigl( \frac
y{|z+i|^2}\Bigr)^s\Bigr)\,.
\end{aligned}\ee
In both estimates $z\rightarrow \p\uhp$ through the region to the left
of $g$ or on~$g$.
\end{lem}
We first apply this result, postponing its proof.

\begin{prop}Let $p\geq 2$. The function $u_\ps$ is an element of
$ \Mf^0_s(\Gm)$ for all $\ps \in Z^1(\Fpt^\tess;\W p s)$.
\end{prop}
\begin{proof}Since $0<\re s<1$, it suffices to show that $u_\ps$ is
bounded on the cuspidal sectors $V_\kp\subset\fd $ for any
$\kp\in \fdcu$. (This follows from the Fourier expansion at the
cusp~$\kp$: see \S\ref{ssect-tM} and equation~\eqref{Fourexp-par}, and
use the asymptotic behavior as $y\rightarrow\infty$ of the special
functions in the expansion.)
Take $\x,\eta\in \cu$ with $\kp$ between $\x$ and $\eta$ for the
positive
(counterclockwise) orientation of $\p\H$ and such that $V_\kp$ is
between the geodesic $g_{\kp,\eta}$ from $\kp$ to~$\eta$ and the
geodesic $g_{\x,\kp}$ from $\x$ to~$\kp$. Let $R_1$, $R_2$, $R_3$
denote the regions to the right of the geodesics $g_{\x,\kp}$,
$g_{\kp,\eta}$, and $g_{\eta,\x}$, respectively, as in the picture
below. We have
\be\label{u3} u = u_{\x,\kp}+ u_{\kp,\eta}+u_{\eta,\x}\,. \ee
By Lemma~\ref{lemxieta}, $u_{\x,\kp}$ is bounded on the complement of
$R_1$ (and hence also on $V_\kp$), $u_{\kp,\eta}$ is bounded on the
complement of $R_2$ (and hence also on $V_\kp$), and $u_{\x,\eta}$ is
bounded on the complement of $R_3$ (and hence also on $V_\kp$, since
$V_\kp\cap R_3$ is compact).
\end{proof}
\[\setlength{\unitlength}{1.2cm}
\begin{picture}(3,3)(-1.5,-2)
\circth
\put(1.5,0){\circle*{.1}}
\put(1.6,-.1){$\kp$}
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\put(-.24,-.1){$V_\kp$}
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\qbezier(1.06,1.06)(.5,.3)(1.5,0)
\qbezier(1.06,-1.06)(.5,-.3)(1.5,0)
\put(-1,0){$R_3$}
\put(1.05,-.5){$R_1$}
\put(1.05,.3){$R_2$}
\end{picture}\]

\begin{prop}\label{prop-ipc}The map $[\ps]\mapsto u_\ps$ from
$\Hpar^1(\Gm;\W p s)$ to $\E_s^\Gm$ is injective for $p\geq 3$.
\end{prop}
\begin{proof}Suppose that $u_\ps=0$. Let $A$ be a path from the cusp
$\x$ to the cusp $\eta$ as in the definition of $u_{\x,\eta}$
in~\eqref{uxieta}. Lemma~\ref{lemxieta} shows that
\be \label{upe1}
\pi i\, u_{\x,\eta} (z) \= \tilde\ps(A)(z) + \oh\Bigl( \Bigl(\frac
y{|z+i|^2}\Bigr)^{s+p-2}\Bigr)
\ee
for $z$ on or on the left of the geodesic from $\x$ to~$\eta$, and
also that $\pi i\, u_{\x,\eta}(z) = - \pi i \,u_{\eta,\x}(z)$
satisfies \eqref{upe1} on the right of that geodesic. Hence
\eqref{upe1} holds for all $z\in \uhp$.

Since $\tilde\ps(A)\in \G_s^p$, there is a continuous function $B$ on
a neighborhood of~$\proj\RR$ in~$\proj\CC$ such that
\[ \tilde\ps(A)(z) \= \Bigl( \frac y{|z+i|^2}\Bigr)^s \, B(z) +
\oh\Bigl( \Bigl(\frac y{|z+i|^2}\Bigr)^{s+1}\Bigr)\qquad
(z\rightarrow\p\uhp)\,. \]
With \eqref{upe1}, and with the assumption $p\geq 3$, we get the same
estimate for $\pi i\, u_{\x,\eta}(z)$. Lemma~4.4 in~\cite{BLZ} tells
us that elements of $\E_s$ satisfying an estimate of this type
vanish. We conclude that $u_{\x,\eta}=0$ for all cusps $\x$
and~$\eta$.

Now we have by \eqref{upe1} that $\tilde \ps(A)\in \N_s^{p-2}$, so
$\ps(A)$ is $0$ in $\W{p-2}s$. Then $\ps(A)=0$ because
$\W p s \rightarrow \W{p-2}s$ is injective
(since $\W p s \cong \V p s \subset \V{p-2}s \cong \W {p-2}s$).


Take a cusp $\x$ as base point. The group cocycle
$\gm\mapsto \ps(C_{\gm^{-1} \x,\x})$ with $C_{\gm^{-1}\x,\x}$ a path
in $\ZZ[X_1^\tess]$ from $\gm^{-1}\x$ to~$\x$ is zero. Hence the
class $[\ps]\in \Hpar^1(\G,m;\W p s)$ is the trivial cohomology
class.
\end{proof}

We summarize the results in the following proposition:
\begin{prop}\label{prop-dcohc}Suppose that $\Gm$ has cusps. For all
$p,q\in \NN$, $q\geq p \geq 3$, the following diagram is commutative
and all arrows are isomorphisms.
\[ \xymatrix{& \Mf^0_s(\Gm)\ar@<.5ex>[dl]^{\bold q}
\\
\Hpar^1(\W\om s,\W\sacp s)\ar[r] \ar@<.5ex>[ur]^{\al_s^\om} &
\Hpar^1(\W\sacp s) \ar[r] & \Hpar(\W q s) \ar[r] \ar[ul]^{\al_s^q} &
\Hpar(\W p s)
\ar[ull]^{\al_s^p} } \]
\end{prop}
\medskip


It remains to prove the estimate~\eqref{xieta}.
\begin{proof}[Proof of Lemma~\ref{lemxieta}] Since
$\tilde \ps(C)\in \G _s ^p$, the second estimate in~\eqref{xieta}
follows from the first.

By conjugation and symmetry it suffices to consider $\x=\infty$,
$\eta=0$. Then $g=i\,\RR_+\,$. We can assume that $i\in \fd$.
Changing $A$ means adding an element of $\N_s^p$ to the sum, and does
not influence the estimate. So we can assume that $A$ runs from
$\infty$ to~$0$ through the left half of the upper half-plane. It now
suffices to estimate for $z\in \uhp$ with $\re z\geq 0$ the sum
\[ \sum_\gm h(\gm z) \;\ll\; \sum_\gm \Bigl( \frac {\im\gm z}{|\gm
z+i|^2} \Bigr)^{s+p}\,,\]
where $\gm\in \Gm$ is such that $\gm^{-1}\fd$ is in the left half
of~$\uhp$ and where $h=d\tilde\ps(\fd) \in \N_s^p$.

For any $\gm=\matc abcd$ in the sum we have $\re \gm^{-1}i <0$, so
$ab+cd>0$. Moreover, $\infty$ is a cusp of~$\Gm$, so $\im\gm^{-1}i$
stays under a bound $B_\infty$, depending only on the group~$\Gm$ and
the cusp~$\infty$. Hence $a^2+c^2 \geq \frac 1{B_\infty}$. For
$\re z\geq 0$, $|z|\geq 1$:
\beqabclbl{ee}
\be
\begin{aligned}
\frac{\im \gm z}{|\gm z+i|^2} \left/\frac y{|z+i|^2} \right. &\=
\frac{|z|^2+2y+1}{(a^2+c^2)|z|^2+2(ab+cd)x+2y+b^2+d^2}\\
&\;\leq\; \frac{|z|^2+2|z|+1}{(a^2+c^2)|z|^2} \;\leq\; 4 B_\infty\,.
\end{aligned}
\ee
We apply the reflection $z\mapsto 1/\bar z$ to conclude that there is
a bound $B_0>0$ such that for $\re z\geq 0$, $|z|\leq 1$:
\be \frac{\im \gm z}{|\gm z+i|^2}\left/\frac y{|z+i|^2} \right.
\;\leq\; 4B_0\,.\ee\eeqabc


In the proof of Lemma~\ref{lem-C-dt} we saw that the number of
$\Gm$-translates of $\fd$ that intersect the closed disk $D_k$
around~$i$ with radius $k$ is $\oh(e^k)$ as $k\rightarrow\infty$. We
use this to estimate the size of the set $F_k$ of $\gm$ occurring in
the sum such that $\gm^{-1}\fd$ intersects $D_k$ but does not
intersect $D_{k-1}$. We use the estimates in~\eqref{ee} for the sum
up to $k=k_0-1$, with $k_0$ to be chosen later.
\[ \sum_{k=0}^{k_0-1} \sum_{\gm\in F_k } h(\gm z)\;\ll\;
\sum_{k=0}^{k_0-1} e^k\, \Bigl(\frac y{|z+i|^2}\Bigr)^{s+p} \;\ll\;
e^{k_0}\, \Bigl(\frac y{|z+i|^2}\Bigr)^{s+p}\,. \]




For the tail of the series, we employ another estimate. Applying the
triangle inequality $\dist(\gm z,i) + \dist(z,i)
\geq \dist(\gm^{-1}i,i)$, and using
$\frac y{|z+i|^2}\sim e^{-\dist(z,i)}$ as $z\rightarrow\p\uhp$, we
find
\[ \frac{\im \gm z}{|\gm z+i|^2} \; \frac y{|z+i|^2} \;\sim\;
e^{-\dist(\gm z,i)-\dist(z,i) } \;\leq\; e^{-\dist(\gm^{-1}i,i)}
\;\ll\; \frac{\im\gm^{-1}i}{|\gm^{-1}i+i|^2}\,, \]
and if $\gm^{-1}\fd$ has distance at least $k-1$ to~$i$, then
\[ \frac{\im \gm^{-1} i}{|\gm^{-1}i+i|^2} \;\ll\; e^{-k}\,.\]
This leads to
\begin{align*}
\sum_{k\geq k_0} & \sum_{\gm\in F_k} h(\gm z)
\;\ll\; \sum_{k\geq k_0} e^k \, \Bigl( \frac{\im
\gm^{-1}i}{|\gm^{-1}i+i|^2}\Bigr)^{s+p} \, \Bigl( \frac
y{|z+i|^2}\Bigr)^{-s-p}\\
&\;\ll\; \sum_{k\geq k_0} e^{(1-s-p)k} \, \Bigl( \frac
y{|z+i|^2}\Bigr)^{-s-p} \;\ll\; e^{(1-s-p)k_0}\, \Bigl( \frac
y{|z+i|^2}\Bigr)^{-s-p}\,.
\end{align*}

Combining both estimates we obtain for the total sum:
\[ \sum_\gm h(\gm z) \;\ll\; e^{k_0-(\re s+p)\dist(z,i)} + e^{(1-\re s
-p)k_0+(\re s+p)\dist(z,i)}\,. \]
We take $k_0=\bigl[ 2\dist(z,i)\bigr]$, and obtain
\[ \;\ll\; e^{(2-\re s-p)\dist(z,i)}\,, \]
which gives the desired estimate of the sum.
\end{proof}



\subsubsection{Injectivity of the map from differentiable cohomology
to invariant eigenfunctions, cocompact case}\label{ssect-dc-cc}


 We can also prove the injectivity of the map from $\W p s$-valued
 cohomology classes to invariant eigenfunctions in the cocompact case,
now even for $p\geq2$, but the proof in the absence of cusps is
harder, because the chain $A$ used in \eqref{uxieta} must be replaced
by an infinite path. We will show:
\begin{prop}\label{prop-dcohcc}Let $\Gm$ be cocompact, and let
$p,\,q\in \NN$, $q\geq p\geq 2$. Then the following diagram is
commutative and all arrows are isomorphisms.
\[ \xymatrix{& \E_s^\Gm\ar@<.5ex>[dl]^{\bold q}
\\
H^1(\W\om s)\ar[r] \ar@<.5ex>[ur]^{\al_s^\om}
& H(\W q s) \ar[r] \ar[u]^{\al_s^q} & H(\W p s)
\ar[ul]^{\al_s^p} } \]
\end{prop}

Only the injectivity of $[\ps] \mapsto u_\ps$ from
$\Hpar^1(\Gm;\W p s)$ to $\E_s^\Gm$ remains to be proved. In the
non-cocompact case we used the decomposition \eqref{ups-decomp} to
write the eigenform $u_\ps$ as a sum of two pieces associated to
paths between cusps. Now there are no edges in $X_1^\tess$ that go to
points of the boundary $\p\H$ and we will use paths between
\emph{hyperbolic} points instead. To do this we will extend a lift
$\tilde\ps\in C^1(\Fpt^\tess;\G^p_s)$ of $\ps$ to a function defined
on infinite paths going from a point $P\in X_0^\tess\subset\H$ to a
hyperbolic fixed point on~$\p\H$, like we did in~\S\ref{ssect-injcc}
in the analytic case, using a one-sided average. However, because our
knowledge of the behavior of the extension near the hyperbolic fixed
point is incomplete, we are forced to perform some complicated
estimates.


The proof will consist of five steps, of which steps a) and~b) were
not needed for groups with cusps, and step~d) requires more work than
in the previous subsection.
\begin{enumerate}
\item[a)] Choice of paths from points of $X_0^\tess$ to hyperbolic
fixed points.
\item[b)] Extension of $\tilde\ps$ to these paths.
\item[c)] Definition of $u_{\tilde\ps}(\x_1,\x_2)$ for hyperbolic
fixed points $\x_1$ and~$\x_2$.
\item[d)] Proof that if $u_\ps=0$ then $u_{\tilde\ps}(\x_1,\x_2)=0$.
\item[e)] Proof that if $u_\ps=0$ then $[\ps]$ is the trivial
cohomology class.
\end{enumerate}

\sssect{Step a)} We assume that the tesselation $\tess$ of type \Fcoh\
is based on a Dirichlet fundamental domain~$\fd$. We choose once and
for all a $\Gm$-orbit $H$ of hyperbolic fixed points of~$\Gm$, and
consider only hyperbolic points belonging to this orbit. For each
$\x\in H$ we denote by $\eta_\x\in \Gm$ the generator of $\Gm_\x$
that has $\x$ as its repelling fixed point.

In the proof of Proposition~\ref{prop-H1P0} we extended a cocycle on
$X_0^\tess \times X_0^\tess$ to a cocycle on
$\left( X_0^\tess \cup H\right)\times \left( X_0^\tess \cup H\right)$.
Here we work with a cochain $\tilde\ps$, and have to deal with actual
paths between points of $ X_0^\tess \cup H$. We choose for each
$\x\in H$ and each $P\in X_0^\tess\subset \H$ an infinite chain
$p(P,\x)$ of elements of $X_1^\tess$, describing a path from $P$ to
$\x$. We require:
\begin{enumerate}
\item[(i)] \emph{$\Gm$-equivariance}, {\sl i.e.},
$\gm p(P,\x)=p(\gm P, \gm \x)$ for all $\gm\in \Gm$,
\item[(ii)] $p(P,\x)$ has no self-intersection,
\item[(iii)] for $P,Q\in X_0^\tess$ and $\x\in H$ the difference
$ p(Q,\x) - p(P,\x)$ is finite, {\sl i.e.}, it is in $\ZZ[X_1^\tess]$
\end{enumerate}


\twocolwithpictr{\quad\ To see that such a choice is possible we start
with one $\x\in H$, and abbreviate $\eta_\x=\eta$. Let $s$ be the
geodesic connecting $\x$ with the other fixed point $\x'$ of~$\eta$.
(In the sketch on the right in the upper half-plane model we place
$\x$ at $0\in\proj\RR$ and $\x'$ at $\infty\in \proj\RR$.)
We choose a point $a$ on $s$ and cover the segment of $s$ between $a$
and $\eta^{-1}a$ with the finitely many translates $\gm \fd$ such
that the closure $\overline{\gm \fd}$ contains the segment between
these two points. }{\setlength\unitlength{.7cm}
\begin{picture}(4,6)(-2,0)
\put(-2,0){\line(1,0){4}}
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\put(0,5){\circle*{.1}}
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\put(.4,2.3){$\eta^{-1}a$}
\put(.1,4.8){$a$}
\put(.1,1){$s$}
\put(0,-.4){$\xi$}
\put(.1,5.9){$\xi'$}
% geodesic
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%
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\qbezier(-.25,2.3)(0,2.4)(.25,2.4)
\qbezier(-.25,2.7)(0,2.75)(.25,2.75)
\end{picture} }\vskip.4ex

Next we take the infinite cover of $s$ consisting of all translates by
$\eta^n$, $n\in \ZZ$, of the cover of the segment between $a$
and~$\eta^{-1}a$.

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\begin{picture}(4,6)(-2,0)
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\put(.4,2.3){$\eta^{-1}a$}
\put(.1,4.8){$a$}
\put(.1,1){$s$}
\put(-1.3,4){$C$}
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%
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%
\put(-.4,3.1){\line(0,1){.7}}
%
\qbezier(-.4,3.1)(-.6,3)(-.8,2.9)
\qbezier(-.25,2.7)(-.3,2.7)(-.5,2.6)
\qbezier(-.5,2.6)(-.7,2.8)(-.8,2.9)
%
\put(-.25,2.3){\line(0,1){.4}}
\qbezier(-.25,2.3)(-.2,2.2)(-.2,1.9)
\put(-.2,1.65){\line(0,1){.35}}
\qbezier(-.2,1.65)(-.3,1.5)(-.4,1.45)
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\qbezier(-.25,1.3)(-.35,1.4)(-.4,1.45)
\put(-.125,1.15){\line(0,1){.2}}
\put(-.1,.8){\line(0,1){.15}}
\put(-.07,.5){\line(0,1){.15}}
\put(-.04,.2){\line(0,1){.15}}
\end{picture} }{\quad\ The union of all these translates is a
connected simply connected region in~$\H$. The boundary of this
region consists of two components. We take the component on the right
 of the geodesic with respect to the direction from $\x'$ to~$\x$.

 This component can be described as an infinite chain of the form
$C=\sum_{i=-\infty}^\infty \e_i\, x_i$ with $x_i\in X_1^\tess$ and
$\e_i\in \{1,-1\}$. The chain is $\eta$-invariant: there exists
$n\geq 1$ such that $\eta x_i = x_{i+n}$. It has no
self-intersections. }\vskip.4ex

 We use this chain to form infinite paths from each $P\in X_0^\tess$
 to~$\x$. Through each $P\in X_0^\tess$ there is a unique geodesic
$g_P$ that intersects $s$ orthogonally. We denote the intersection
point by $x_P$. If $P$ happens to be a point of~$s$ we take $x_P=P$.

\twocolwithpictr{\quad\ First we consider those points
$P\in X_0^\tess$ for which $x_p$ is between $a$ and $\eta^{-1}a$, or
is equal to~$a$. For each of these points $P$ we choose a chain
$r\in \ZZ[X_1^\tess]$ describing a path from $P$ to a point $Q$ in
$C \cap X_0^\tess$. This can be done in such a way that the path
corresponding to $r$ intersects the path given by $C$ only in~$Q$.
Then we choose $p(P,\x)$ as the sum of $r$ and the part of the chain
$C$ describing a path from $Q$ to~$\x$.}{
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\put(0,4.4){\circle*{.1}}
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%
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\put(-.07,.5){\line(0,1){.15}}
\put(-.04,.2){\line(0,1){.15}}
\end{picture} }\vskip.4ex

The path $x(P,\x)$ that we have constructed has no self-intersection.
If $P,R\in X_0^\tess$ satisfy $x_P, x_R\in [a,\eta^{-1}a)$ then
$p(R,\x)$ and $p(P,\x)$ form the same path, except for an initial
part. So requirement~(iii) is satisfied. Thus we have completed the
choice of paths going to the fixed $\xi\in H$.

 he condition on the intersection points $x_P$ implies that the set
$H\times X_0^\tess$ is freely generated by the $(\x,P)$ for which we
have defined $p(P,\x)$. We put $p(\Gm P,\gm\x) = \gm p(P,\x)$ for
$\gm\in \Gm$ to define $p$ on $H\times X_0^\tess$. This choice
clearly satisfies requirements (i)
and~(ii). For requirement~(iii) we observe that $p(\eta^{-n}P,\x)$ and
$p(P,\x)$ differ only in an initial part.\smallskip

In the sequel we assume that $p(P,\x)$ for $\x\in H$ and
$P\in X_0^\tess$ has been chosen such that requirements (i)--(iii)
hold.



\sssect{Step b)}The lift $\tilde\ps  \in C^1(\Fpt^\tess;\G_ s^ p)$ of
the given cocycle $\ps\in Z^1(\Fpt^\tess;\W ps )$ is defined on paths
in $\ZZ[X_1^\tess]$. For $P\in X_0^\tess$ and $\x\in H$ we have
arranged that $p(P,\x) - \eta_\x^{-1}p(P,\x) \in \ZZ[X_1^\tess]$. So
$\tilde \ps\bigl( p(P,\xi)-\eta_\x^{-1}p(P,\x)\bigr)$ is an element
of $\G_s^p$. We use the one-sided average $\av{\eta_\x}^+$ discussed
in \S\ref{ssect-avhyp} to define:
\be F_{\tilde\ps}(P,\x) \= \av {\eta_\x}^+ \tilde\ps\bigl(
(1-\eta_\z^{-1})p(P,\x)\bigr) \;\in \; \G^p_s\bigl( \p\H\sm
\{\x\}\bigr)\,. \ee
The dependence of $F_{\tilde\ps}(P,\x)$ on the choice of the path
$p(P,\x)$ is not visible in the notation. We think of
$F_{\tilde\ps}(P,\x)$ as $\tilde\ps$ evaluated on the infinite path
$p(P,\x)$.

Near points of $\p \H \sm \{\x\}$ we have good information on the
behavior of $F_{\tilde\ps}(P,\x)$. It is locally (in the disk model)
of the form
$w\mapsto \bigl( 1-\nobreak|w|^2\bigr)^s \cdot (\text{analytic})$,
as defined in~\S\ref{ssect-bj}. The next lemma gives information
concerning the behavior near the point~$\x$. We formulate it in the
upper half-plane model, with $\x$ at position zero, and use polar
coordinates $\uhp \ni z=\rho e^{i\ph}$ ($\rho>0$, $\ph\in (0,\pi)$).

\begin{lem}\label{lem-Fpsi-est}Let $g\in G$ such that $g \cdot 0=\x$,
in the upper half-plane model, and
$g\eta_\x g^{-1} = \matc{\sqrt t}00{1/\sqrt t}$ with $t>1$. Then we
have, uniformly for $0<\ph<\pi$:
\begin{alignat*}2 F_{\tilde\ps}(P,\x)(g \cdot
\rho e^{i\phi} )
&\;\ll\; 1&\qquad&(\rho\downarrow0)\,,\\
\p_\rho F_{\tilde \ps}(P,\x)(g \cdot \rho e^{i\phi} ) &\;\ll\;
\rho^{-1}&&(\rho\downarrow0)\,.
\end{alignat*}
\end{lem}
\begin{proof}This is a statement concerning
$H(\rho e^{i\phi})=\sum_{n\geq 0} h(t^n x)$ for some $h\in \G^p_s$.
There are $R_\infty > R_0 >0$ such that
\[ h(\rho e^{i\phi}) \= \begin{cases} \rho ^s
(\sin \phi)^s a_0(\rho \cos\phi,\rho \sin \phi)&\text{ for }0<\rho
\leq R_0\,,\\
\oh(1)&\text{ for } R_0\leq \rho \leq R_\infty\,,\\
\rho ^{-s}(\sin\phi)^s a_\infty(-\rho ^{-1}\cos\phi, \rho ^{-1}\sin
\phi)&\text{ for } \rho \geq R_\infty\,,
\end{cases}
\]
with $C^p$-functions $a_0 $ and $a_\infty$ on a neighborhood of $0$ in
$\RR^2$ containing a disk of radius $R_0$, respectively
$R_\infty^{-1}$. In the intermediate region we also have
$\p_\rho  h(\rho e^{i\phi})=\oh(1)$.

For $\rho <R_0$ we split up the sum at
$B=- \frac{\log( \rho/R_0)  }{\log t}$ and
$A=-\frac{\log (\rho / R_\infty)}{\log t}$. The sum over
$0\leq n < B$ contributes
\begin{align*}
\text{to }H(\rho e^{i\phi}):\quad \sum_{0\leq n<B}& t^{ns}\rho ^s(\sin
\phi)^s a_0(t^n \rho \cos\phi,t^n \rho \sin \phi)
\;\ll\; \rho ^s(\sin\phi)^s t^{Bs}\\
&\;\ll\; \rho ^s (\sin \phi)^s \rho ^{-s} \;\ll\; 1\,,\\
\text{to }\p_\rho H(\rho e^{i\phi}):\quad\;\ll\; \sum_{0\leq n < B}&
\rho ^s(\sin\phi)^s \left( t^{ns}\rho ^{-1}+ t^{n(s+1)} \right)
\;\ll\; \rho ^{-1}\,.
\end{align*}


The sum over $n\geq A$ contributes
\begin{align*}
\text{to }H(\rho e^{i\phi}):\quad \sum_{n\geq A}& t^{-ns}\rho
^{-s}(\sin \phi)^s a_\infty(-t^{-n}\rho ^{-1}\cos\phi,t^{-n}\rho
^{-1}\sin\phi)\\
&\qquad\hbox{} \;\ll\; \rho ^s \rho ^{-s} \;\ll\; 1\,,\\
\text{to }\p_\rho H(\rho e^{i\phi}):\quad\sum_{n\geq A}&
t^{-n(s+1)}\rho ^{-s}(\sin\phi)^s \left( s \rho ^{-1} \oh(1) + \rho
^{-2} \oh(1)
\right)
\;\ll\; \rho ^{-1}\,.
\end{align*}


The region $B\leq n<A$ contributes $\oh\bigl(A-B\bigr) = \oh(1)$ to
$H(\rho e^{i\phi})$, and to $\p_\rho  H(\rho e^{i\phi})$
\[ \sum_{B\leq n < A} t^n \oh(1) \;\ll\; t^A \;\ll\; \rho ^{-1}\,. \]
On this region we do not obtain a factor $\rho^s(\sin\ph)^s$. So we
have to be content with $\oh(1)$ and $\oh(\rho^{-1})$ as the final
estimates.
\end{proof}

Let $p(\dv,\dv)$ and $\hat p(\dv,\dv)$ denote two choices of paths,
both satisfying the requirements. Then $\hat p(P,\x)-p(P,\x)$ can be
written as an infinite sum $\sum_{\gm\in \Gm} m_\gm \, (\gm^{-1}\fd)$
of elements of $X_2^\tess$, with $m_\gm=\pm 1$ if $\gm^{-1}\fd$ is
between both paths, with the choice of the sign depending on the
winding number of $\hat p(P,\x)-p(P,\x)$ around $\gm^{-1}\fd$, and
$m_\gm=0$ for all other $\gm^{-1}\fd$. For the influence of the
choice of the path on $F_{\tilde\ps}(P,\x)$ we would like to get an
estimate of the sum
\be\label{se}\sum_\gm m_\gm \, d\tilde\ps(\gm^{-1}\fd)\,.\ee



For later use we formulate this more generally. We need estimates for
absolutely converging sums over $\gm$ such that $\gm^{-1}\fd$ is
contained in a set $X$. Near pieces of the boundary away from the
closure of $X$ in $\proj\CC$ the estimates are better. In the proof
of Lemma~\ref{lem-uxy0} we will need also an estimate for derivatives
of the sum.

 For convenience we use in the following lemma the disk model
 $\H=\disk$, with coordinates $w=re^{i\th}$, $0\leq r<1$ and
$\th\in \RR/2\pi\ZZ$.

\begin{lem}\label{lem-tps}
Let $X$ be a union of $\Gm$-translates of $\fd$. Denote by $\bar X$
the closure of $X$ in $\disk\cup \p\disk$. Let $h\in \N^p_s$,
$p=2,3,\ldots$, let $\{\e_\gm\;:\; \Gm\}$ be a bounded set of complex
numbers, and put
\[ H \= \sum_{\gm\in \Gm\,,\; \gm^{-1}\fd \subset X} \e_\gm\,
h\,|\,\gm \,.\]
Then for any $w\in \disk$ with hyperbolic distance $\dist(w,X)$ to $X$
at least equal to $R\geq 0$
\be\label{dist-e} H(w) \,\ll\, e^{-(s+p-1)R}\,,\quad
(1-r^2)\, \p_r H(re^{i\th}) ,\; (1-r^2)\, \p_\th H(re^{i\th})\,\ll\,
e^{-(s+p-1)R}\,. \qedhere\ee
\end{lem}
The differential operators operators $(1-\nobreak r^2)\, \p_r$ and
$r^{-1}\,(1-\nobreak r^2)\,\p_\th$ are natural in the coordinates $r$
and~$\th$, since they are $\Gm$-equivariant up to a factor of
absolute value one.

\begin{proof}In fact, we will prove the estimates \eqref{dist-e} for
the functions obtained by replacing all terms in the defining sums by
their absolute values.

Near the boundary, $h(re^{i\th}) = (1-\nobreak r^2)^s\, a(re^{i\th})$
with a $C^p$-function $a$ on a neighborhood of the boundary,
vanishing up to order~$p$ on the boundary. Hence
$h(re^{i\th}) = \oh\bigl( (1-\nobreak r^2)^{s+p}\bigr)$ and
$\p_r h(re^{i\th})$ and $\p_\th h(re^{i\th})$ are
$ \oh \bigl( (1-\nobreak r^2)^{s+p-1}\bigr)$. Since $h$ is a
$C^2$-function, we can use these estimate everywhere on $\disk$. To
estimate $\p_r\sum h\circ\gm$ and $\p_\th \sum h\circ\gm$ we use
$\sum \bigl|\frac{d\gm w}{dw}\,h_r(\gm w)\bigr|$ and
$\sum \bigl|\frac{d\gm w}{dw}\,h_\th (\gm w)\bigr|$, and note that
$\bigl|\frac{d\gm w}{dw} \bigr|= \frac{1-|\gm w|^2}{1-|w|^2}$.
(We use that $h_r$ and $h_\th$ are linear combinations of $h_w$ and
$h_{\bar w}$ with bounded coefficients.) So for $H$ and for its
derivatives we have to deal with sums of the type
\be\label{sq} \sum_{\gm\in \Gm,\, \gm^{-1}\fd \subset X} \bigl( 1-|\gm
w|^2\bigr)^{s+q}\,,\ee
with $q=p$ or $q=p-1$, and have to add a factor
$\bigl(1-\nobreak r^2\bigr)$ for the derivatives.

We can assume that $0\in \fd$. For all $\gm\in \Gm$
\[ 1-|\gm w|^2 \;\ll\; e^{-\dist(\gm w,0) }\=
e^{-\dist(w,\gm^{-1}0)}\;\leq\; e^{-\dist(w,\gm^{-1}\fd)}\,.\]
 An area consideration shows that the number of $\gm\in \Gm$ such that
 $\dist(w,\gm^{-1}\fd)\leq R$ is $\oh\bigl(e^R\bigr)$. Hence the sum
 in~\eqref{sq} satisfies
 \[ \;\ll\; \sum_{l\geq 1} e^{\dist(w,X)+l}\,
 e^{-(s+q)(l+\dist(w,X))}\;\ll\; e^{-(s+q-1)\dist(w,X)}\,.\]
This gives~\eqref{dist-e}.
\end{proof}

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{Lemma~\ref{lem-tps} has as a direct consequence that if $I$ is a
closed cyclic interval in $\uc=\p\disk$ contained in the open set
$\p\disk \sm \bar X$, then as $r\uparrow 1$, uniform in
$e^{i\th}\in I$:
\be\label{1mr}
\begin{aligned}
H(re^{i\th}) &\;\ll\; (1-r^2)^{s+p-1}\,,\\
\p_r H(re^{i\th}) &\;\ll\; (1-r^2)^{s+p-2}\,.
\end{aligned}
\ee
To see this we simply observe that
$e^{-\dist(w,X)} = \oh((1-\nobreak r^2)$ uniformly, because the
nearest point of $X$ to any point near $I$ lies in a fixed compact
subset of~$X$. }\vskip.5ex

If we have two choices $\hat p(P,\x)$ and $p(P,\x)$ for the path from
$P$ to~$\x$ we get for the difference of the corresponding values
$\hat F_{\tilde\ps}(P,\x)$ and $F_{\tilde\ps}(P,\x)$ the expression
\[ \sum_{\gm \in \Gm} m_\gm \, d\tilde\ps(\gm^{-1}\fd) \= \sum_{\gm
\in \Gm} m_\gm \, d\tilde\ps(\fd)|\gm \]
with $m_\gm$ as in~\eqref{se}. We have $d\tilde\ps(\fd)\in \N^ p_ s$,
and the estimate \eqref{1mr} shows that this sum is estimated by
$\oh\bigl( (1-r^2)^{s+p-1}\bigr)$ near points of
$\p\disk \setminus \{\x\}=:I$. The restriction morphism $
\rho_s: \W p s (I)\rightarrow \V p s(I)$ in \S\ref{ssect-bj} is given
by $\rho_s f( \x) 
= \lim_{r\uparrow 1} \bigl( (1-r^2))^{-s} \, f(r e^{i\th}) \bigr)$ on
the class in $\W p s(I)$ represented by $f\in \G^p _s (I)$. Thus, we
obtain the following result:
\begin{lem}\label{lem-cH}The element
$\rho_s F_{\tilde\ps}(P,\x)\in \V p s(\p\disk\sm\{\x\})$ does not
depend on the choice of $p(P,\x)$, and satisfies
\be \rho_s F_{\tilde\ps}(\gm^{-1}P,\gm^{-1}\x) \= \bigl( \rho_s
F_{\tilde\ps}(P,\x)
\bigr)|\gm\qquad (\gm\in \Gm)\,,\ee
and
\be\label{Pxicoc} \rho_s F_{\tilde\ps}(P_1,\x) \= \rho_s
F_{\tilde\ps}(P,\x)+\ps(r)
\ee
for $P_1,P\in X_0^\tess$, $\x\in H$ and $r\in \ZZ[X_1^\tess]$
corresponding to a path from $P_1$ to~$P$.
\end{lem}

Thus we get a cocycle $c$ on
$(X_0^\tess\cup\nobreak H) \times\allowbreak (X_0^\tess\cup\nobreak H)$
with values in a $\Gm$-module containing $\V p s$ in which
singularities at points in $H$ are allowed. The restriction of $c$ to
$X_0^\tess\times X_0^\tess$ is related to
$\ps\in Z^1(\Fpt^\tess;\W p s)$ by $c(P,Q) = \rho_s \ps(p)$, where
$p\in \ZZ[X_1^\tess]$ is a path from $P$ to~$Q$. For $P\in X_0^\tess$
and $\x\in H$ we have $c(P,\x)=\rho_s F_{\tilde\ps}(P,\x)$.



\sssect{Step c)}Let $\x_1,\x_2 \in H$, and choose $P\in X_0^\tess$.
Motivated by \eqref{uxieta} we might consider
\smallskip
\twocolwithpictr{
\begin{align*}
&F_{\tilde\ps}(P,\x_2)-F_{\tilde\ps}(P,\x_1)\\
&\quad\hbox{} + \sum_{\gm\in \Gm,\, \gm^{-1}\fd \subset X} d\tilde
\ps(\fd)|\gm\,,
\end{align*}
where $X$ is in the region indicated on the left. However, this sketch
is misleading, because the paths $p(P,\x_1)$ and $p(P,\x_2)$ may
intersect each other. }{
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\qbezier(0,-.4)(.1,-.3)(.3,.2)
\end{picture}
}\vskip.8ex

We need to work with a sum $\sum_\gm m_\gm \, d\tilde\ps(\fd)|\gm$,
where $m_\gm$ is the winding number around $\gm^{-1}\fd$ of the
closed path $-p(P_,\x_2)+p(P,\x_1)$ from $\x_2$ via $P$ to $\x_1$ and
then back to $\x_2$ along the positively oriented arc of $\p\disk$
from $\x_1$ to~$\x_2$.
(In the sketch above, where $p(P,\x_1)$ and $p(P,\x_2)$ do not
intersect each other except in $P$, we have $m_\gm=1$ if
$\gm^{-1}\fd \subset X$ and $m_\gm=0$ otherwise.)
We define
\be \label{ux1x2def}
u_{\tilde\ps}(\x_1,\x_2)
\= \frac1{\pi i}\,\Bigl( F_{\tilde\ps}(P,\x_2)-F_{\tilde\ps}(P,\x_1)
+ \sum_{\gm\in \Gm} m_\gm\, d\tilde\ps(\fd)|\gm \Bigr)\,. \ee
There is a more general representation
\be \label{u0}
u_{\tilde\ps}(\x_1,\x_2) \= \frac1{\pi i} \,\Bigl( \tilde \ps(A) +
\sum_{\gm\in \Gm}m_\gm\, d\tilde\ps(\fd)|\gm\Bigr)\,,\ee
\twocolwithpictr {where $A$ is any infinite path given by an infinite
chain of the form $-p(Q_1,\x_1)+r+p(Q_2,\x_2)$, where
$Q_1,Q_2\in X_0^\tess$, where $r\in \ZZ[X_1^\tess]$ corresponds to a
path from $Q_1$ to $Q_2$, and where $m_\gm$ is the winding number
around $\gm^{-1}\fd$ of the closed path consisting of $-A$ and the
positively oriented arc in $\p\disk$ from $\x_1$ to~$\x_2$.}{
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\begin{lem}\label{lem-ux1x2E}
We have $u_{\tilde\ps}(\x_1,\x_2)\in \E_s$ and
\be\label{u-decomp} u_{\tilde\ps}(\x_1,\x_2)+ u_{\tilde\ps}(\x_2,\x_1)
\= u_\ps\,.\ee
\end{lem}
\begin{proof}The second assertion follows directly from \eqref{upsavp}
and~\eqref{u0}.

To see that $u_{\tilde\ps}(\x_1,\x_2)$ is in $\E_s$ we want to apply
the same reasoning as in Proposition~\ref{prop-upsp}. We need a
sequence $A_l$ of infinite paths from $\x_1$ to~$\x_2$ that have
distance at least $l$ to a fixed point~$P_0$. To construct such a
sequence we consider first the sequences $(\eta_1^{-n}P_0)$ and
$(\eta_2^{-n}P_0)$ tending to $\x_1$ and $\x_2$, respectively. Put
$p_n = \eta_1^{-n} p(P_0,\x_1) - \eta_1^{-n-1} p(P_0,\x_2)$. This is
a finite path from $\eta_1^{-n}P_0$ to $\eta_1^{-n-1}P_0$, and
$F_{\tilde\ps}(P_0,\x_1) $ is equal to
$ \sum_{n\geq 0} \tilde\ps(p_0)|\eta_1^n$.
(The infinite chain $\sum_{n\geq 0}p_n$ describes an infinite path
from $P_0$ to~$\x_1$, that will in general be different from
$p(P_0,\x_1)$.) For each $n$ the maximum of the distance between two
points of $p_n$ does not depend on~$n$. The sequence of finite paths
$q_n= \eta_2^{-n} p(P_0,\x_2) - \eta_2^{-n-1} p(P_0,\x_2)$ has
similar properties.


 For each $l\geq 1$ there is by Lemma~\ref{lem-C-dt} a cycle $Z_l$
 going around $\disk$ once in the positive direction in the region
with hyperbolic distance at least $l$ to~$P_0$. There are points
$P_l\in X_0^\tess$ in the intersection of $Z_l$ with
$\sum_{n\geq 0} p_n$, and $Q_l\in X_0^\tess$ in the intersection of
$Z_l$ with the infinite path $\sum_{n\geq 0} q_n$. Take $n_1$ such
that $P_l$ is on the finite path $p_{n_1}$, and $n_2$ such that $Q_l$
is on the path $a_{n_2}$. We form the chain $A_l$ as the sum of the
following chains:
\[\begin{array}{rclll}
&&&\text{from}&\text{to}\\
a_1&=&- \sum_{n\geq n_1}p_n&\x_1&\eta_1^{-n_1}P_0\,,\\
a_2&& \text{ along edges of }p_{n_1}&\eta_1^{-n_1}P_0 & P_l\,,\\
a_3&& \text{ along edges of } C_l&P_l&Q_l\,,\\
a_4&&\text{ along edges of } q_{n_2}&Q_l&\eta_2^{-n_2}P_0\,,\\
a_5&=& \sum_{n\geq n_2}q_n & \eta_2^{-n_2}P_0&\x_2\,.
\end{array}
\]

To apply the method used in the proof of Proposition~\ref{prop-upsp}
we estimate $(\Dt-\nobreak \ld_s) \tilde \ps(a_j)$ for each of these
paths. We have $(\Dt-\nobreak \ld_s) \tilde \ps(a_3) 
= \oh(e^l) \, \oh(e^{-(s+p)l}) = o(1)$. The paths $a_2$ and $a_4$
consist of finitely many edges, and have distance at least
$l- \oh(1)$ to $P_0$. This leads to
$(\Dt-\nobreak \ld_s) \tilde \ps(a_i)=o(1)$ for $i=2,4$. The path
$q_{n_2}$ has distance at least $l-\oh(1)$ to~$P_0$, and there exists
a factor $\al>0$ such that $\eta_2^{-n} q_{n_2}$ has distance at
least $\al n + l - \oh(1)$ to~$P_0$. Hence
$(\Dt-\nobreak \ld_s) \tilde \ps(a_5 ) 
\ll \sum_{n\geq 0} \oh\bigl(e^{-(s+p)(\al n + l ) }\bigr) = o(1)$,
and similarly for $a_1$.


Now we can proceed as in the proof of Proposition~\ref{prop-upsp}.
\end{proof}



\sssect{Step d)} We now show:
\begin{lem}\label{lem-uxy0}If $u_\ps=0$, then
$u_{\tilde\ps}(\x_1,\x_2) = 0$ for all $\x_1,\x_2 \in H$.
\end{lem}
\begin{proof}
Since $u_\ps=0$, equations \eqref{u-decomp} and \eqref{ux1x2def} give
\begin{align}
\label{u0rel}
 \pi i \,u_{\tilde\ps}(\x_1,\x_2) &\= F_{\tilde\ps}(P,\x_2)-
 F_{\tilde\ps}(P,\x_1)
+ S_{\tilde\ps}(P,\x_1,\x_2)\\
\nonumber
&\= -\pi i \,u_{\tilde\ps}(\x_2,\x_1) \=
-F_{\tilde\ps}(P,\x_1)+ F_{\tilde\ps}(P,\x_2) -
S_{\tilde\ps}(P,\x_2,\x_1)\,,
\end{align}
where
\be S_{\tilde \ps}(P,\x,\eta) \= \sum_{\gm\in \Gm } m_\gm \,
d\tilde\ps(\fd)|\gm\,,
\ee
with $m_\gm$ the winding number around $\gm^{-1}\fd$ of the path
consisting of $-p(P,\eta)$, $p(P,\x)$ and the positively oriented arc
in $\p\disk$ from $\x$ to~$\eta$.

Let $w_1\in \disk$. By Lemma~\ref{lem-ux1x2E} and \eqref{[]def} we can
use Theorem~\ref{thm-psC} to write
\be\label{ux12int}
u_{\tilde\ps}(\x_1,\x_2)(w_1)
= -2\pi \int_C \bigl\{ u_{\tilde\ps}(\x_1,\x_2),v\}\,,\ee
where $v(w) = q_s(w,w_1)$ and $C$ is a curve encircling $w_1$ once in
the positive direction.\smallskip


\twocolwithpictr{\pridtc We will choose the path $C$ consisting of
four pieces, two being small arcs $C_1$ and $C_2$ near $\x_1$ and
$\x_2$, and two being arcs $D_1$ and $D_2$ between $\x_1$ and~$\x_2$.
(See diagram.)

\pridtc Near $\x_j$ it is convenient to use the upper half-plane
model, writing $\x_j = k_j \cdot 0$, with
$k_j \in \mathrm{PSO}(2)\subset G=\PSL_2(\RR)$. }{
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\put(-.33,-1.46){\circle*{.1}}
\put(-.4,-1.9){$\x_2$}
\put(-.3,-.95){$C_2$}
\put(-1.3,-.15){$D_1$}
\put(.87,-.15){$D_2$}
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\end{picture}}\vskip.8ex

We choose the $C_j$ depending on two small positive parameters $\e$
and $\dt$, as indicated in the following sketch of $k_j^{-1}\, C_j$
in the upper half-plane:
\[
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\qbezier(1,0)(1,.16)(.89,.27)
\put(1.1,.06){$\dt$}
\qbezier(-1,0)(-1,.16)(-.89,.27)
\put(-1.33,.06){$\dt$}
\put(1.7,2){$k_j^{-1}C_j$}
\put(-.95,2){$|z|=\e$}
\put(2.4,.95){$k_j^{-1}z_j$}
\put(2.4,.683){\circle*{.1}}
\put(-3.2,.95){$k_j^{-1}z_j'$}
\put(-2.4,.683){\circle*{.1}}
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\]
The points $z_j$ and $z_j'$, where the arc $C_j$ meets $D_1$ and
$D_2$, correspond to $w_j=\frac{z_j-i}{z_j+i}$ and
$w_j'=\frac{z_j'-i}{z_j'+i}$ in the disk model. The absolute value
$b:=|w_j|=|w_j'|$
(near $1$) is related to $\e$ and $\dt$
(near $0$)
by an explicit formula (namely
$1-b^2=\frac{4\e\sin\dt}{1 +2\e\sin \dt+\e^2}$). The arcs $D_j$ are
of the form $w=b e^{i\th}$ where $\th$ runs through an interval $I_j$
in $\RR/2\pi\ZZ$. We must show that each of the four contributions to
\eqref{ux12int} tends to zero as $\dt\rightarrow0$ and then
$\e\rightarrow0$.

We begin with $C_j$. In the upper half-plane model we use for $z$ near
$\x_j$ the coordinate $\rho e^{i\ph}=k_j^{-1} z\in \uhp$, for which
\[ \{u,v\}\= \frac1\rho\,\bigl( vu_\ph-uv_\ph\bigr)\,
d\rho+\rho\,\bigl( u v_\rho-vu_\rho\bigr)\, d\ph\,,\]
and hence simply
$\{u,v\} = \e\,\bigl(uv_\rho\nobreak-vu_\rho)\bigr)\,d\ph$ on $C_j$,
where $\rho=\e$ is constant.

We have $F_{\tilde\ps}(P,\x_j)=\oh(1)$ and
$\p_\rho F_{\tilde\ps}(P,\x_j) = \oh(\e^{-1})$ according to
Lemma~\ref{lem-Fpsi-est}. The corresponding term for
$F_{\tilde\ps}(P,\x_{j'})$, with $j'\neq j$, is smaller, since
$F_{\tilde\ps}(\x_1,\x_2)\in \G^p_s(J)$ for an interval $J$
containing~$\x_j$; the derivative is $\oh(\rho^{s-1})$.
Lemma~\ref{lem-tps} and its consequence \eqref{1mr} applied to
$S_{\tilde\ps}(P,\x_1,\x_2) = - S_{\tilde\ps}(P,\x_2,\x_1)$ show that
$S_{\tilde\ps}(P,\x_1,\x_2)$ is bounded near $\x_j$, and that near
$\x_j$ the derivatives with respect to $r$ and $\th$ are estimated by
$\oh\bigl( (1-\nobreak r^2)^{-1}\bigr) = \oh(\e ^{-1})$. The
derivative with respect to $\rho$ can be expressed in these
derivatives with bounded coefficients. Finally, the functions $v$ and
$v_\rho$ for $v=q_s(\dv,w_1)\in \W p s$ are $\oh(\e^s)$ and
$\oh(\e^{s-1})$ respectively. All this leads to the following
estimate of the integral over~$C_j$:
\[ \int_{\pi-\dt}^\dt \e \Bigl( \oh(1)\, \oh(\e^{1-s}) + \oh(\e^s)\,
\oh(\e^{-1})
\Bigr)\, d\phi \= \oh(\e^s)\,.\]
This estimate is uniform in $\dt\in (0,\pi/2)$.
\smallskip

Now we turn to $D_j$. In the disk model with coordinate
$w=re^{i\th} \in \disk$ we have
\[ \{u,v\}\= \frac 1r\,\bigl( vu_\th-uv_\th\bigr)\, dr + r\, \bigl(
uv_r-vu_r\bigr)\, d\th \,, \]
and hence $\{u,v\}= b\, \bigl( uv_r-vu_r)\, d\th$ on $D_j$, where
$r=b$ is constant. Since $I_j$ has length at most $2\pi = \oh(1)$,
the contribution of the integral over $D_j$ is bounded by a multiple
of the maximum of $|uv_r|+|vu_r|$ over the arc~$D_j$.

We use that $F_{\tilde\ps}(P,\x_j)\in \G^p_s(I)$ and that $v$
represents an element of $\W p s$ to get from
$F_{\tilde\ps}(P,\x_j)(re^{i\th}) =
(1-\nobreak r^2)^s a(r,\th)$ and
$v(re^{i\th}) = (1-\nobreak r^2)^s b(r,\th)$ the contribution
\[ r(1-r^2)^{2s} \bigl( a(r,\th)b_r(r,\th) - a_r(r,\th)b(r,\th)
\bigr)\, d\th\,.\]
This contributes $\oh_\e\bigl( (1-\nobreak b^2)^{2s} \bigr)$ to the
integral. (We note that the implicit constant depends on the
interval, hence on~$\e$.)
By Lemma~\ref{lem-tps} the contribution of
$S_{\tilde\ps}(P,\x_1,\x_2)$ or $-S_{\tilde\ps}(P,\x_2,\x_1)$ is
$\oh_\e\left(
(1-\nobreak b^2)^{2s+p-2}\right)$. Since $p\geq 2$, the final estimate
of the integral over $D_j$ is $\oh_\e\bigl((1-b^2)^{2s}\bigr)$.
\smallskip

We have arrived at
\[ u_{\tilde\ps}(\x_1,\x_2) \= \oh(\e^s)+ \oh_\e\bigl(
(1-b^2)^{2s}\bigr)\,,\]
with the first term uniform in $\dt$. Letting $\dt\downarrow 0$, and
hence $b\uparrow1$, we arrive at the bound $\oh(\e^s)$ for all
positive~$\e$. Hence $u_{\tilde\ps}(\x_1,\x_2)=0$.
\end{proof}


\sssect{Step e)}If $u_\ps=0$, then
$u_{\tilde\ps}(\x_1,\x_2)=u_{\tilde\ps}(\x_2,\x_1)=0$ by step~d).
Lemma~\ref{lem-uxy0} and \eqref{u0rel} give for $P\in X_0^\tess$ and
$\x_1,\x_2\in H$:
\be F_{\tilde\ps}(P,\x_2) - F_{\tilde \ps}(P,\x_1) \=
S_{\tilde\ps}(P,\x_2,\x_1)
\= - S_{\tilde\ps}(P,\x_1,\x_2)
\,. \ee
The cocycle $c$ on $(X_0^\tess\cup\nobreak H)
\times\allowbreak(X_0^\tess\cup\nobreak H) $ introduced after
Lemma~\ref{lem-cH} satisfies
$c(\x_1,\x_2)=\rho_s\bigl( F_{\tilde\ps}(P,\x_2) 
- \nobreak F_{\tilde\ps}(P,\x_1)\bigr)$. We have
$F_{\tilde\ps}(P,\x_j)(w)= A_j(w)\, (1-\nobreak|w|^2)^s$ where $A_j$
is $p$ times differentiable on a neighborhood $\Om_j$ of
$\uc\setminus\nobreak\{\x_j\}$ in $\proj\CC$. Estimate~\eqref{1mr}
shows that $S_{\tilde\ps}(P,\x_1,\x_2)$ is
$o\bigl(1-\nobreak|w|^2\bigr)^{p-1}$ near closed intervals in
$(\x_2,\x_1)_c$ and, with reversed roles of $\x_1$ and $\x_2$, also
near $(\x_1,\x_2)_c$. So $A_1-A_2$ is in
$C^{p-1}\bigl(\Om_1\cap \Om_2\bigr)$. We glue $A_1$ and $A_2$ to
obtain $f\in \G_s^{p-1}$ that represents $c(P,\x_j)$ on
$\uc\setminus\{\x_j\}$ for $j=1,2$. The sheaf $\W {p-1}s$ has no
sections with support consisting of one point, so
$F_{\tilde\ps}(P,\x_1)$ and $F_{\tilde\ps}(P,\x_2)$ are in
$\W {p-1}s$ and and their difference is represented by an element of
$\N _s^{p-1}$. The proof can now be completed in the same way as in
the last part of the proof of Proposition~\ref{prop-ipc}.


\subsection{Smooth parabolic cohomology}\label{sect-smcoh}We combine
the results on $\V p s$-valued cohomology of the previous section to
show that $\Hpar^1(\Gm;\V\infty s)\cong \Mf^0_s(\Gm)$. In this way,
we complete the proofs of Theorems~\ref{thmA} and~\ref{thmB}, and
  conclude that the main theorem of~\cite{LZ} extends to three times
differentiable period functions.


\begin{thm}\label{thm-sm}Let $0<\re s<1$. Let $\Gm$ be a cofinite
discrete subgroup of $G=\PSL_2(\RR)$.

If $\Gm$ is cocompact then $\E^\Gm_s$ is isomorphic to
$H^1(\Gm;\V\infty s)$ and to $H^1(\Gm;\V p s)$ for all $p\in \NN$,
$p\geq 2$.

If $\Gm$ has cusps, then $\Mf^0_s(\Gm)$ is isomorphic to
$\Hpar^1(\Gm;\V\infty s)$ and to $\Hpar^1(\Gm;\V p s)$ for all
$p\in \NN$, $p\geq 3$.
\end{thm}
\begin{proof} With the restriction map $\rho_s$ we obtain from
Propositions~\ref{prop-dcohc} and \ref{prop-dcohcc}, in the case that
$\Gm$ has cusps and $p\geq 3$
\[ \xymatrix{\Mf^0_s(\Gm) \ar[r]^(.43){\cong}
& \Hpar^1(\Gm;\V\sacp s) \ar[r] \ar@/^4mm/[rr]^\cong
& \Hpar^1(\Gm;\V\infty s) \ar[r]
& \Hpar^1(\Gm;\V p s)
} ,\]
and if $\Gm$ is cocompact and $p\geq 2$
\[\xymatrix{ \E_s^\Gm \ar[r]^(.35)\cong & H^1(\Gm;\V\om s) \ar[r]
\ar@/^4mm/[rr]^\cong
& H^1(\Gm;\V\infty s) \ar[r]
& H^1(\Gm;\V p s)
}, \]
where in both cases the composition is an isomorphism. So it suffices
to show that the natural maps
$\Hpar^1(\Gm;\V\infty s)\lra \Hpar^1(\Gm;\V p s)$ and
$H^1(\Gm;\V\infty s)\lra H^1(\Gm;\V p s)$ are injective for all $p$.


We show this in the standard model of cohomology. Suppose that for
$\ps\in Z^1(\Gm;\V\infty s)$ there exists $a_p\in \V p s$ such that
$\ps_\gm = a_p\,|\,(\gm-\nobreak 1)$ for all $\gm\in \Gm$. Since all
arrows in the diagrams in Propositions~\ref{prop-dcohc} and
\ref{prop-dcohcc} are isomorphisms, there are also $a_q\in \V q s$
such that $\ps_\gm=a_q\,|\,(\gm-\nobreak 1)
$ for all $\gm$. We take a hyperbolic $\gm\in\Gm$, and conclude from
Proposition~\ref{prop-hyp-inv} that all $a_q$ coincide, and hence
give elements of $\V\infty s$.
\end{proof}

As a consequence of \eqref{Hpc} we obtain for the modular group the
following extension of the main theorem in \cite{LZ} on period
functions and Maass cusp forms:
\begin{thm}\label{thm-pfsm}\indind{period function}{thm-pfsm} Let
$0<\re s<1$, and $p\in \NN$, $p\geq 3$.
\[ \Mf_s^0(\Gmod) \;\cong\: \pf_s(\RR_+)^0_\om \=
\pf_s(\RR_+)^0_\infty \= \pf_s(\RR_+)^0_p\,.\]
\end{thm}
So thrice differentiable functions on $(0,\infty)$ that satisfy
\[f(x) \= f(x+\nobreak 1)
\,+\, (x+\nobreak 1)^{-2s}\,f\Bigl( \frac x{x+1} \Bigr)^s\]
and the estimates $\ps(x) = \oh(1)$ as $x\downarrow0$ and
$\ps(x) =\oh(x^{-2s})$ as $x\rightarrow\infty$ automatically are real
analytic, and occur as the period function of a Maass cusp form.


\subsubsection{Recapitulation of the proof of
Theorem~\ref{thmA}}\label{recapA}
The definition in \eqref{rdef} induces an injective (Proposition
\ref{prop-rinj}) map $\bold r : \E_s^\Gm\lra H^1(\Gm;\V\om s)$. The
isomorphism $\V\om s \cong \W\om s$ (in \S\ref{ssect-tPm}) and
Theorem~\ref{thm-cc-upsi} give a left inverse
$\al_s^\om\circ \tpoiss s$ of~$\bold r$, with
\[ \al : [\ps] \mapsto \bb(s)\, u_{[\ps]}\,,\]
with $\bb(s)$ as in~\eqref{btd}. The injectivity of
$\al_s^\om\circ \tpoiss s$ follows from
Proposition~\ref{prop-av-inj}, Proposition~\ref{prop-H1P0} and the
exact sequence \eqref{P-longexact}.

Proposition~\ref{prop-alp} extends $\al_s^\om$ to
$\al_s^p :H^1(\Gm;\W p s)\lra \E^\Gm_s$, in a compatible way for
different values of~$p$. Theorem~\ref{thm-sm} shows that these
extensions are injective, and also determine an isomorphism
$H^1(\Gm;\V\infty s) \lra \E^\Gm_s$.



\subsubsection{Recapitulation of the proof of
Theorem~\ref{thmB}}\label{recapB}The injective map $\bold r$ in
 Proposition~\ref{prop-rinj} from $ \Mf^0_s(\Gm) $ to
 $ H^1(\Gm;\V\om s)$ has its image in the subspace
$ \Hpar^1(\Gm;\V\om s,\V\sa s)
$, according to Proposition~\ref{prop-im-coh}.
Proposition~\ref{prop-M0M1} shows that
\[\bold r : \Mf^0_s(\Gm) \lra \Hpar^1(\Gm;\V\om s,\V\sa s)\]
is an isomorphism. Propositions \ref{prop-W} and the fact that
$\V\sa s$ is locally defined (Definition~\ref{def-locdef})
imply that
\[ \Hpar^1(\Gm;\V\om s,\V\sa s) \;\cong\; \Hpar^1(\Gm;\V\sa s)\,.\]
Finally, Theorem~\ref{thm-sm} gives
\[ \Mf_s^0(\Gm) \;\cong\; \Hpar^1(\Gm;\V\infty s)\,.\]














\skipbetweenchapters
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%   Chapter 6   Petersson product and cup product         %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\sect{Distribution cohomology and Petersson product}\label{ch-VI}




In the previous chapters we have considered the relation between
automorphic forms and cohomology groups with values in principal
series spaces consisting of functions, possibly with singularities.
In this chapter we turn to cohomology with values in the spaces
$\V{-\infty}s$ and $\V{-\om}s$ of distribution and hyperfunction
vectors in the principal series. For groups with cusps,
$H^1(\Gm;\V{-\infty}s)$ is isomorphic to $\Mf^0_s(\Gm)$, by a result
of Bunke and Olbrich, \cite{BO98}.\footnote{See the footnote in
\S\ref{sect-co-Maf-cc} on p.~\pageref{sect-co-Maf-cc}. } The natural
map from $H^1(\Gm;\V{-\infty}s)$ to $ H^1(\Gm;\V{-\om}s)$ turns out
to be the zero map if $\Gm$ has cusps. This contrasts with the
cocompact case, where Bunke and Olbrich, \cite{BO95}, have shown that
$H^1(\Gm;\V{-\infty}s)= H^1(\Gm;\V{-\om}s)$.

The Petersson scalar product can be transformed into a bilinear form
on the space $\Mf^0_s(\Gm)$. The isomorphisms with cohomology groups
transform this bilinear form into a duality between
$\Hpar^1(\Gm;\V\infty s)$ and $H^1(\Gm;\V{-\infty}{1-s})$. We will
show in Section~\ref{sect-dual} that this bilinear form coincides, up
to a multiple, with that given by the cup product, which we have to
adapt to parabolic cohomology if the discrete group has cusps.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Distribution cohomology}\label{sect-distrcoh}

The obvious way to obtain a map from Maass forms to distribution
cohomology uses the natural homomorphism associated to
$\V\infty s\lra \V{-\infty}s$. In \S\ref{ssect-vanim} we see that
this leads to the zero map. In \S\ref{ssect-cf-dc} we discuss another
map from Maass forms to distribution cohomology, which gives an
isomorphism on cusps forms.




\subsubsection{Vanishing image in distribution and hyperfunction
cohomology}\label{ssect-vanim}

The injective map $\bold r$ in Proposition~\ref{prop-rinj} can be
followed by the natural map to $ H^1(\Gm;\V\infty s)$. The
composition is injective for cusp forms:
\[ \xymatrix{ \Mf_s^0(\Gm)\ar[r]^(.4){\bold r}_(.4){\cong}
\ar@{^{(}->}[d]
& \Hpar^1(\Gm;\V\om s,\V\sa s)
\ar[r]_(.55)\cong \ar@{^{(}->}[d]
& \Hpar^1(\Gm;\V\infty s) \ar@{^{(}->}[d]\\
\E_s^\Gm \ar@{^{(}->}[r]^(.4){\bold r}
& H^1(\Gm;\V\om s) \ar[r]
& H^1(\Gm;\V\infty s)
} \]
The inclusions $\V\infty s\subset\V{-\infty}s \subset\V{-\om}s$ give
further natural homomorphisms
\be\label{nathom}
H^1(\Gm;\V\infty s)\lra H^1(\Gm;\V{-\infty}s)
\lra H^1(\Gm;\V{-\om}s)\,. \ee
\begin{prop}The image of $\bold r \,\E_s^\Gm$ under the natural maps
in \eqref{nathom} vanishes in $H^1(\Gm;\V{-\om}s)$, and the image of
$\bold r\, \Mf_s(\Gm)$ vanishes in $H^1(\Gm;\V{-\infty}s)$.
\end{prop}
\begin{proof}Let $u\in \E_s^\Gm$. In \S\ref{ssect-Maf-coh} we have
seen that the class of $\bold r u$ is represented by a
$\V\om s$-valued cocycle $\ps_\gm=g|_{2s}\,(\gm-\nobreak1)$, where
$g\in \HHH$ is a representative of the hyperfunction
$\poiss s^{-1} u$. Hence the image of $\bold r u$ in
$H^1(\Gm;\V{-\om}s)$ is zero. If $u\in \Mf_s(\Gm)$ then $u$ has
polynomial growth. Theorem~\ref{thmLew} implies that
$\poiss s ^{-1}u\in \V{-\infty}s$. Hence $\bold r u$ is zero in
$H^1(\Gm;\V{-\infty}s)$.
\end{proof}




\subsubsection{From cusp forms to distribution
cohomology}\label{ssect-cf-dc}
In \S2.4 of \cite{BLZ} we have considered two spaces of
\indlbl{soef}{second order eigenfunctions}\emph{second order
eigenfunctions:}
\be\label{E'def}
\begin{aligned}
\E_s' &\= \ker\bigl( (\Dt-\ld_s)^2: C^\infty(\H)\lra C^\infty(\H)
\bigr)\,,\\
(\E^{-\infty}_s)'&\= \bigl\{ f\in \E_s'\;:\; f \text{ has polynomial
growth}\bigr\}\,.
\end{aligned}
\ee
The space of eigenfunctions of polynomial growth is equal to
$\E^{-\infty}_s=\poiss s \V{-\infty}s$
(Theorem~\ref{thmLew}).

Propositions 2.6 and~2.7 in \cite{BLZ} show that the spaces of second
order eigenfunctions fit into the following exact sequences:
\be\label{E'ex}
\begin{array}{ccccccccc}
0&\lra&\E_s&\lra&\E_s'& \stackrel{\Dt-\ld_s}\lra&\E_s&\lra&0
\smallskip\\
&&\cup&&\cup&&\cup \\
0&\lra&\E_s^{-\infty}&\lra&(\E_s')^{-\infty}&
\stackrel{\Dt-\ld_s}\lra&\E_s^{-\infty}&\lra&0
\end{array}
\ee
The second of these exact sequences leads to a non-zero map from cusp
forms to distribution cohomology. Indeed, define
\indlbl{bbd}{$\bold b$}$\bold b : \Mf_s(\Gm) \lra H^1(\Gm;\V{-\infty}s)$
as the composition
\[ \Mf_s(\Gm) \= (\E^{-\infty}_s)^\Gm \= H^0(\Gm;\E^{-\infty}_s) \lra
H^1(\Gm;\E^{-\infty}_s) \;\cong\; H^1(\Gm;\V{-\infty}s)\,,\]
of the connecting homomorphism in the long exact sequence associated
to the second row in \eqref{E'ex} and the isomorphism provided by the
Poisson transformation.

If $\Gm$ has cusps and $u\in \Mf_{s_0}(\Gm)$ is the value $u_{s_0}$ of
a holomorphic family $s\mapsto u_s\in \Mf_s(\Gm)$, then a
$\Gm$-invariant lift $\tilde u_{s_0}\in
(\E'_s)^{-\infty}$ can be obtained by differentiation with respect
to~$s$, as in the proofs of Propositions 2.6 and~2.7 in~\cite{BLZ}.
So $\bold b$ vanishes on Eisenstein series. Bunke and Olbrich have
shown in \cite{BO98}, Proposition~8.1:
\begin{prop}Let $0<\re s<1$. The map $\bold b : \Mf_s^0(\Gm)
\lra H^1(\Gm;\V{-\infty}s)$ is an isomorphism.
\end{prop}
Their proof uses an exact sequence
\[ 0 \lra \E^{-\infty}_s\lra \Ccal^{-\infty} \stackrel{\Dt-\ld_s}\lra
\Ccal^{-\infty}\lra 0\,,\]
where $\Ccal^{-\infty}$ consists of the $f\in C^\infty(\uhp)$ such
that $\p_z^n\p_{\bar z}^m f$ has polynomial growth for all
$n,m\in \NN$.

In the cocompact case, this result amounts to
$\E^\Gm_s \cong H^1(\Gm;\V{-\infty}s)$. In \cite{BO95}, Bunke and
Olbrich have shown that
\be \E^\Gm_s \;\cong\; H^1(\Gm;\V{-\infty}s)
\= H^1(\Gm;\V{-\om}s)\,.
\ee

If $\Gm$ has cusps and $s\neq\frac12$, then all elements $u\in \E_s$
occur in holomorphic families, as we have seen in part~ii) of
Theorem~\ref{thm-hex}. Differentiation with respect to~$s$ gives for
$u\in \E_s^\Gm$ a $\Gm$-invariant lift in $(\E'_s)^\Gm$. Hence the
image of $\bold b u $ in $H^1(\Gm;\V{-\om}s)$ vanishes:
\begin{prop}Let $0<\re s<1$, $s\neq\frac12$. For groups with cusps the
composition
\[ \Mf_s(\Gm) \stackrel{\bold b}\lra H^1(\Gm;\V{-\infty}s) \lra
H^1(\Gm;\V{-\om}s)\]
is the zero map.
\end{prop}


\subsection{Duality}\label{sect-dual}In this final section we give a
cohomological description of the Petersson scalar product.


\subsubsection{Petersson scalar product}The \emph{Petersson scalar
product} on the space of cusp forms $\Mf^0_s(\Gm)$ is given
by\indlbl{Pp}{Petersson product; $(\dv,\dv)$}
\[ (u,v) \= \int_{\Gm\backslash\uhp} u \, \bar v \; d\mu\,.\]
It can be computed by integration over any measurable fundamental
domain, for instance over $\fd$ as chosen in \S\ref{ssect-cc-ex} in
the cocompact case and \S\ref{ssect-res-c} if $\Gm$ has cusps.

Instead of $(\dv,\dv)$ we use the \emph{bilinear Petersson scalar
product}\indlbl{bPp}{bilinear Petersson product;
$\langle\dv,\dv\rangle$}
\be \langle u,v \rangle \= \int_{\Gm\backslash\uhp} u \, v\;
d\mu\qquad\bigl(u,v\in \Mf^0_s(\Gm)\bigr)\,.
\ee
Since $\Mf^0_s(\Gm) \neq \{0\}$ only if $s\in
\bigl(\frac12+i\RR\bigr)\cup(0,1)$, the space $\Mf^0_s(\Gm)$ is
invariant under conjugation. Hence $(u,v)=\langle u,\bar v \rangle$
for all $u,v\in \Mf^0_s(\Gm)$ for all~$s$ with $0<\re s<1$.


\subsubsection{Cup product}\label{ssect-cp}
The cup product in cohomology can be described with any augmentation
preserving chain map $\Fpt \lra \Fpt\otimes \Fpt$ of projective
resolutions, called a diagonal approximation, which gives an
isomorphism in cohomology. In the standard model of group cohomology
one may use the diagonal approximation described in \cite{Brn},
Chap.~V, \S1. This leads to a linear map\indlbl{cup}{cup product;
$\cup$}
\[\cup: H^i(\Gm;V) \otimes H^j(\Gm;W) \lra H^{i+j}(\Gm;V\otimes
W)\,.\]
For $i=j=1$ this is induced on $1$-cocycles by $(b\cup c)(x) = -
(b\otimes c)(\Dt_2 x))$, where $\Dt_\pnt$ denotes a diagonal
approximation. For $i=j=1$, the cup product ends up in
$H^2(\Gm;V\otimes W)$, which is isomorphic to $(V\otimes W)_\Gm$ in
the cocompact case. For groups with cusps, the second cohomology
groups vanish, and we need parabolic cohomology to have a reasonable
cup product. Applied in the cocompact case, we get back the usual cup
product.

For our purpose it suffices to consider a resolution $\Fpt^\tess$
based on a tesselation $\tess$ of type~\Mcoh, as discussed in
\S\ref{ssect-cc-ex} and~\S\ref{ssect-res-c}. The tensor product
$G_{\!\pnt}= \Fpt^\tess\otimes \Fpt^\tess$ with $G_i =
\bigoplus_{a=0}^i F_a^\tess\otimes F^\tess_{i-a}$ gives a resolution
of $\QQ$. The boundary maps are determined by
\indlbl{cpd}{$\p_j$}$\p_{i+j} x\otimes y =(\p_i x)\otimes y
+(-1)^i x \otimes(\p_j y)$ for $x\in F_i^\tess$ and $y\in F_j^\tess$.
The augmentation is given by
\indlbl{augmcp}{$\e$}$\e \left( P\otimes Q \right)=1$ for
$P,Q\in X_0^\tess$. With minimal sets $B_i \subset X_i^\tess$ of
generators of $F_i^\tess$, the sets
\[ \bigcup_{a=0}^i \left\{ x \otimes\gm^{-1}y\;:\; x\in B_a,\, y\in
B_{i-a},\, \gm\in \Gm_y\backslash \Gm\right\}\]
generate $G_i$ over $\QQ[\Gm]$, and form a basis if $i\geq 1$. For
$\kp\in \cu$:
\[ G_0^{\pi_\kp} \= \QQ\cdot \kp \otimes\kp\,.\]
These are the sole generators with an infinite isotropy subgroup. The
conclusion is that $G_\pnt$ is a parabolic resolution as defined
in~\S\ref{ssect-par-coh}.


\sssect{Explicit basis elements}For the sequel we need an explicit
description of generating elements of $F^\tess_i$ for the tesselation
$\tess$ of type \Mcoh. The Dirichlet fundamental domain $\fd$
underlying the tesselation has the following boundary:
\be \p\fd \= \sum_{e\in E} e\,|\,(1-\gm_e^{-1}) +
\begin{cases}
0&\text{ (cocompact)}\,,\\
\sum_{\kp\in \fdcu} e_\kp \,|\,(1-\pi_\kp^{-1})&\text{ (not
cocompact)}\,.
\end{cases}
\ee
If $\Gm$ is cocompact, then $\fdcu$ is empty.

We introduce some notation, illustrated in Figure~\ref{fig-me} for the
modular group. Define vertices \indlbl{PQe}{$P_e$; $Q_e$}$P_e$ and
$Q_e$ of $\fd_\hy$ (or of $\fd$ if $\Gm$ is cocompact) such that
$e=\allowbreak e(P_e,Q_e)$, and also vertices $P_\kp$ of $\fd_\hy$
such that $e_\kp=e(P_\kp,\kp)$ if $\Gm$ has cusps. Note that $Q_e$ may
occur as $P_{e'}$ for some $e'\in E$.
\begin{figure}[ht]
\[
\setlength{\unitlength}{2.8cm}
\begin{picture}(1.7,2.3)(-.95,.5)
\put(0,1){\circle*{.07}}
\put(-.5,.866){\circle*{.07}}
\put(.5,.866){\circle*{.07}}
\put(-.5,2.1){\circle*{.07}}
\put(.5,2.1){\circle*{.07}}
\put(.55,1.4){$a$}
\put(.55,2.3){$e_\infty$}
\put(.55,2.05){$P_\infty=Q_a$}
\put(.55,.8){$P_a=S P_b$}
\put(-.3,.85){$b$}
\put(-1.15,2.15){$T^{-1}P_\infty=$}
\put(-.9,1.95){$T^{-1}Q_a$}
\put(-.9,1){$P_b=$}
\put(-1,.8){$T^{-1}P_a$}
\put(-.1,.85){$Q_b=$}
\put(-.1,.7){$SQ_b$}
\put(.05,1.3){$P_0$}
\put(-.05,2.17){$f_\infty$}
\thicklines
\put(-.5,.866){\line(0,1){1.7}}
\put(.5,.866){\line(0,1){1.7}}
\qbezier(-.5,.866)(-.3,1)(0,1)
\qbezier(.5,.866)(.3,1)(0,1)
\put(-.5,2.1){\line(1,0){1}}
\put(0,1.3){\circle*{.07}}
%\put(.5,1.4){\circle*{.07}}
%\put(-.5,1.4){\circle*{.07}}
\put(0,1){\line(0,1){.3}}
%\qbezier(0,1.3)(.3,1.4)(.5,1.4)
%\qbezier(0,1.3)(-.3,1.4)(-.5,1.4)
\qbezier(0,1.3)(.25,1.85)(.5,2.1)
\qbezier(0,1.3)(.25,1.2)(.5,.866)
\qbezier(0,1.3)(-.25,1.85)(-.5,2.1)
\qbezier(0,1.3)(-.25,1.2)(-.5,.866)
\put(.5,2.1){\vector(0,1){.3}}
\put(.5,1.2){\vector(0,1){.3}}
\put(-.3,.955){\vector(4,1){.2}}
\put(.23,2.1){\vector(-1,0){.3}}
\end{picture}
\]
\caption{The points $P_e$ and $Q_e$ in the standard fundamental domain
of the modular group.}\label{fig-me}
\end{figure}

So $F_0^\tess$ is generated by $P_0$, the $P_e$ and $Q_e$ with
$e\in E$, the $P_\kp$ and $\kp$, with $\kp\in \fdcu$. We use the
$\QQ[\Gm]$-basis of $F_1^\tess$ consisting of the $e\in E$, the edges
$e(P_0,R)$ where $R$ runs through the set of vertices
$\{P_e, Q_e\}\cup \{P_\kp,\pi_\kp^{-1}P_\kp\}$, the $e_\kp=e(P_\kp,\kp)$
and the $f_\kp=e(P_\kp,\pi_\kp^{-1}P_\kp)$ with $\kp\in \fdcu$. A
$\QQ[\Gm]$-basis of $F_2^\tess$ consists of the polygons in
$X_2^\tess$ contained in~$\fd$. For the tesselation $\tess$ of type
\Mcoh\ these polygons are triangles. Their sum represents the
fundamental class. \indind{fundamental class}{fuclH} Denoting by
$\Dt(A,B,C)$ the triangle in $X_2^\tess$ with boundary
$e_{A,B}+e_{B,C}+e_{C,A}$ we have:
\begin{align}\label{fuclH}
(\fd) &\= \sum_{e\in E} \Bigl( \Dt(P_0,P_e,Q_e) +
\Dt(P_0,\gm_e^{-1}Q_e,\gm_e^{-1}P_e) \Bigr)
\\
\nonumber
&\qquad\hbox{}
+ \sum_{\kp\in \fdcu} \Bigl( \Dt(P_0,P_\kp,\pi_\kp^{-1}P_\kp) + V_\kp
\Bigr)\,,
\end{align}
where $V_\kp = \Dt(\pi_\kp^{-1}P_\kp,P_\kp,\kp)$.

\sssect{Explicit diagonal approximation} The cup product in parabolic
cohomology is induced by any augmentation preserving chain map
$\dt_\pnt:\Fpt^\tess \lra \Fpt^\tess \otimes \Fpt^\tess$. With the
notations just introduced we indicate a special choice that will work
well in connection with the Petersson scalar product.

In dimension~$0$ there is only one sensible choice:
\be \dt_0 (P) \= (P) \otimes (P)\quad (P\in X_0^\tess)\,. \ee
This is continued $\QQ[\Gm]$-linearly, and gives an augmentation
preserving map.

In dimension $1$ we prescribe $\dt_1$ on the basis discussed above.
For each of the basis elements $e_{P,Q}$ we put
\be \dt_1 e_{P,Q} \= P \otimes e_{P,Q}+e_{P,Q}\otimes Q\,. \ee
After $\QQ[\Gm]$-linear extension, $\dt_1$ turns out to be compatible
with the boundary maps: $\p_0 \dt_1=\dt_1\p_1$.

For the basis elements of $F_2^\tess$ we make the choice in
Table~\ref{tab-dt2b}, with $e\in E$, $\kp\in\fdcu$.
\begin{table}[ht]
\[
\renewcommand\arraystretch{1.3}
\begin{array}{|rl|}\hline
\dt_2 \Dt(P_0,P_e,Q_e) &\;=\; P_0\otimes \Dt(P_0,P_e,Q_e) +
e(P_0,P_e)\otimes e \\
&\quad\hbox{}
+ \Dt(P_0,P_e,Q_e)\otimes Q_e
\\ \hline
\dt_2 \Dt(P_0,\gm_e^{-1}Q_e,\gm_e^{-1}P_e)
&\;=\; P_0\otimes\Dt(P_0,\gm_e^{-1}Q_e,\gm_e^{-1}P_e)
\\
&\quad\hbox{}
- e(P_0,\gm_e^{-1}P_e)\otimes e(\gm_e^{-1}P_e,\gm_e^{-1}Q_e) \\
&\quad\hbox{}
+ \Dt(P_0,\gm_e^{-1}Q_e,\gm_e^{-1}P_e)
\otimes \gm_e^{-1}Q_e
\\ \hline
\dt_2 \Dt(P_0,P_\kp,\pi_\kp^{-1}P_\kp)&\;=\;
P_0\otimes\Dt(P_0,P_\kp,\pi_\kp^{-1}P_\kp) + e(P_0,P_\kp)\otimes f_\kp\\
&\quad\hbox{}
+ \Dt(P_0,P_\kp,\pi_\kp^{-1}P_\kp) \otimes \pi_\kp^{-1}P_\kp
\\ \hline
\dt_2 V_\kp &\;=\; P_\kp\otimes V_\kp - f_\kp\otimes \pi_\kp^{-1} e_\kp +
V_\kp \otimes\kp\\
\hline
\end{array}
\]
\caption{Basis elements of $F^\tess_2$, with $e$ running through $E$
and $\kp$ through~$\fdcu$.} \label{tab-dt2b}
\end{table}
It is some work to check that this is compatible with $\p_2$
and~$\p_1$. This choice has the special property
\be \dt_2 F^\tess_2 \subset \left( F_0^\tess \otimes F_2^\tess \right)
\oplus \left( F_1^{\tess,\hy} \otimes F_1^\tess \right) \oplus \left(
F_2^\tess \otimes F_1^\tess \right) \,, \ee
where the first factor of the $(1,1)$-term is~$F_1^{\tess,\hy}$.

\sssect{Cup product}Let $V$ and $W$ be $\QQ[\Gm]$-modules. The cup
product of cocycles $b\in Z^1(\Fpt^{\tess,\hy};V)$ and
$c\in Z^1(\Fpt^\tess;W)$ is computed as
$(b\cup c)(x) = -(b\otimes c)(\dt_2 x)$ for $x\in X_2^\tess$. The
tensor $b\otimes c$ sees only the component of $x$ in
$F_1^{\tess,\hy} \otimes F_1^\tess$. The result represents an element
of $\Hpar^2(\Gm;V \otimes W) $, which does not depend on the choice
of $b$ and~$c$ in their cohomology classes. Thus, we have
obtained\indlbl{cuppar}{cup product for parabolic cohomology; $\cup$}
\be \cup : H^1(\Gm;V) \otimes \Hpar^1(\Gm;W)
\lra \Hpar^2(\Gm;V\otimes W)\,. \ee

By evaluation on the fundamental class we obtain an element of $
\Hpar^2(\Gm;V \otimes W)\cong (V\otimes W)_\Gm $ represented by:
\be
\begin{aligned}
(b\cup c)(\fd) &\= \sum_{e\in E} \biggl(-b(P_0,P_e)\otimes c(e) +
\bigl( b(\gm_eP_0,P_e)\otimes c(e)\bigl)\,|\,\gm_e \biggr)\\
&\qquad\hbox{}
+\sum_{\kp\in \fdcu}\biggl(
- b(P_0,P_\kp) \otimes c(f_\kp) + b(f_\kp)
\otimes c(\pi_\kp^{-1} e_\kp)
\biggr)
\,.
\end{aligned}
\ee

\sssect{Duality}In the special case that there is a $\Gm$-invariant
bilinear form $\langle\dv,\dv\rangle: V \times W \lra \CC$ we have a
linear map $(V\otimes W)_\Gm \lra \CC$. Thus we have a linear form
$H^1(\Gm,V) \otimes \Hpar^1(\Gm;W) \lra \CC$ given by
\[ [b]\otimes [c] \mapsto \Big\langle (b\cup c)([\Gm\backslash\H])
\Big\rangle\,,\]
where we denote by
\indlbl{lin}{$\langle\dv\rangle$}$\langle\dv\rangle$ the linear form
on $V\otimes W$ corresponding to the bilinear form
$\langle\dv,\dv\rangle$. We will use this with $V=\V{-\infty}{1-s}$
and $W=\V\infty s$. See~\S\ref{ssect-mps-bd}.


\subsubsection{Cohomological interpretation of the Petersson scalar
product}
\begin{thm}\label{thm-Ppcp}For $0<\re s<1$ and all cofinite
$\Gm\subset G$, the bilinear Petersson scalar product is given by the
cup product of $\bold b u \in H^1(\Gm;\V{-\infty}{1-s})$ and
$\bold r v \in \Hpar^1(\Gm;\V\infty s)$ evaluated on the fundamental
class:
\[ \langle u,v\rangle \= \frac{-\bb(s)}{2i } \,\left\langle (\bold b u
\cup \bold r v)
([\Gm\backslash\H])\right\rangle \]
for $u,v\in \Mf^0_s(\Gm)$. See \eqref{btd} for the gamma
factor~$\bb(s)$.
\end{thm}
\begin{remarks}
\emph{Remarks.}
\rmk For discrete groups with cusps we understand the cup product in
the parabolic sense of the previous subsection.


\rmk The choice of the spectral parameter $s$ such that
$\ld_s=s(1-\nobreak s)$ is the eigenvalue of $\Dt$ on $u$ and~$v$ is
not visible in the notations $\bold b$ and~$\bold r$. Here it is
important to use opposite choices for the spectral parameter for
$\bold b u$ and $\bold r v$. For the Maass cusp forms the choice does
not matter: $\Mf^0_{1-s}(\Gm)
= \Mf^0_s(\Gm)$.
\end{remarks}

\par\noindent\emph{Proof. } The proof takes the remainder of this
subsection, and consists of three separate steps, which use several
results from the previous chapters. For cocompact $\Gm$ we use
$\fdcu=\emptyset$ throughout the proof.

\sssect{Use of a distribution-valued cocycle} First we use the
description of cohomology with cocycles on the group~$\Gm$.

The map $\bold b$ in \S\ref{ssect-cf-dc} gives rise to
\[\bold b \otimes \mathrm{id} : \Mf^0_{1-s}(\Gm) \otimes \Mf^0_s(\Gm)
\lra H^1(\Gm;\E^{-\infty}_{1-s})\otimes \Mf^0_s(\Gm)\,,\]
where we have used the identification
$\V{-\infty}{1-s} \cong \E^{-\infty}_{1-s}$ by the Poisson
transformation. We define a linear map
$\bold d: Z^1(\Gm;\E_{1-s}) \otimes \Mf_s^0(\Gm) \lra \CC$ by
\be\label{bid-def} \bold d (b\otimes v) \= 2 i \sum_{e\in E'} \int_e [
b_{\gm_e^{-1}},v]\,, \ee
where $E'=E \cup \{ e_\kp \;:\; \kp\in\fdcu\}$ and $\gm_e\in \Gm$ for
$e\in E'$ such that $\p\fd = \sum_{e\in E'} e\,|\,(1-\gm_e)$. For
$e=e_\kp$, $\kp\in \fdcu$, and hence $\gm_e=\pi_\kp$, the convergence of
the integral is assured by the exponential decay of $v$ and its
derivatives, and the polynomial growth of $b_{\gm^{-1}}$.


\begin{lem}\label{lem-tr1}Definition \eqref{bid-def} induces a linear
map
\[\bold d : H^1(\Gm;\E^{-\infty}_{1-s})
\otimes \Mf^0_s(\Gm) \lra\CC\]
such that the following diagram commutes:
\[ \xymatrix{ \Mf_{1-s}^0(\Gm) \otimes \Mf_s^0(\Gm) \ar[d]_{\bold
b\,\otimes\,\mathrm{id}} \ar[rrd]^{\langle\dv,\dv\rangle}
\\
H^1( \Gm;\E_{1-s}^{-\infty}) \otimes \Mf^0_s(\Gm) \ar[rr]^(.6){\bold
d}
&& \CC} \]
\end{lem}
\begin{proof}Let $b\in Z^1(\Gm;\E^{-\infty}_{1-s})$ and $v\in
\Mf_s^0(\Gm)$. To see that $\bold d(b\otimes v)$ does not depend on
the choice of $b$ in its cohomology class, we look at $b=da\in
B^1(\Gm;\E^{-\infty}_{1-s})$.
\begin{align*}
\bold d(b\otimes v) &\= 2 i \sum_{e\in E'} \int_e \left[
a\,|\,\gm_e^{-1}-a,v\right] \= 2 i \sum_{e\in E'} \biggl(
\int_{\gm_e^{-1}e} [a,v] - \int_e[a,v]\biggr)
\\
&\= 2 i\int_{\p\fd}[a,v]\=0\,.
\end{align*}\medskip

Let $u, v\in \Mf^0_s(\Gm)= \Mf^0_{1-s}(\Gm)$. The exactness of the
sequences in \eqref{E'ex} implies that $\bold b u$ can be represented
by a cocycle $\gm\mapsto b_\gm = \tilde u\,|\,(\gm-\nobreak1)$ with
$\tilde u \in
(\E'_{1-s})^{-\infty}$ such that $(\Dt-\nobreak\ld_s) \tilde u=u$.
Thus,
\[ \bold d(\bold b u\otimes v) \= \bold d(b\otimes v) \= 2 i
\sum_{e\in E'}\bigl[ \tilde u\,|\,(\gm_e^{-1}-1),v\bigr] \= 2 i
\int_{\p\fd}\bigl[\tilde u,v\bigr]\,.\]
Since $(\Dt-\ld_s)\tilde u$ may be non-zero, the form $[\tilde u,v]$
is not closed. For cocompact $\Gm$ we obtain with~\eqref{d[]}:
\[ d\bigl( \bold b u \otimes v \bigr) \= 2i \int_{\p\fd}\bigl[ \tilde
u,v \bigr] \= \int_\fd uv\, d\mu \= \langle u,v\rangle\,.\]
If $\Gm$ has cusps, we replace $\fd$ by the truncated fundamental
domain $\fd_a=\fd \cap \uhp_a$ with $a$ large. (See
\S\ref{ssect-res-c} for $\uhp_a$.) The exponential decay of $v $ and
the polynomial growth of $\tilde u$ and their derivatives shows that
for all large values of~$a$
\[ \bold d\bigl( \bold b u \otimes v\bigr) \= 2
i\int_{\p\fd_a}\bigl[\tilde u,v\bigr] + \o(1)
\= \int_{\fd_a} u v\, d\mu + \o(1)
\= \langle u,v\rangle+\o(1)\,. \]
Taking the limit as $a\rightarrow\infty$ we obtain the desired
equality.
\end{proof}


\sssect{Reformulation with a $2$-cocycle}We switch to the description
of cohomology with a tesselation $\tess$ of type \Mcoh, and define
for $b\in Z^1(\Fpt^\tess;\E_{1-s}^{-\infty})$ and
$v\in \Mf_s^0(\Gm)$:
\be\label{om} \om_{P,Q} \= \bigl[ b(P,Q),v\bigr]\qquad(P,Q\in
X_0^\tess)\,.\ee
The map $(P,Q) \mapsto \om_{P,Q}$ defines a $1$-cocycle on
$X_0^\tess\times X_0^\tess$ with values in the smooth closed
differential forms, satisfying $\om_{\gm^{-1}P,\gm^{-1}Q} =
\om_{P,Q}\circ \gm$ for $\gm\in\Gm$. For $P,Q,R\in X_0^\tess$ we put
\be\label{Ccoc-def} C(P,Q,R) \= - 2 i \int_Q^R \om_{P,Q}\,,\ee
where the path of integration from $Q$ to $P$ follows edges
in~$X_1^\tess$.
\begin{lem}\label{lem-tr2}Equation \eqref{Ccoc-def} defines a cocycle
$C\in Z^2(\Fpt^\tess;\CC)$ and induces a linear map
\be \bold a : H^1(\Gm;\E^{-\infty}_{1-s})
\otimes \Mf_s^0(\Gm) \lra \Hpar^2(\Gm;\CC)
\ee
such that the following diagram commutes:
\[ \xymatrix{ H^1( \Gm;\E_{1-s}^{-\infty})
\otimes \Mf^0_s(\Gm)
\ar[rr]^(.7){\bold d} \ar[rrd]^(.7){\bold a}
&& \CC
\\
&
& \Hpar^2( \Gm;\CC) \ar[u]_{\text{eval.~on $(\fd)$}} } \]
\end{lem}
\par\noindent\emph{Proof. }If $P$ or $Q$ is a cusp, the decay
properties used in the proof of Lemma~\ref{lem-tr1} ensure
convergence here as well. The $\Gm$-invariant $\CC$-valued
$2$-cochain $\Dt(P,Q,R)\mapsto C(P,Q,R)$ in $C^2(\Fpt^\tess;\CC)$,
also called~$C$, is automatically in $Z^2(\Fpt^\tess;\CC)$, since
$F_3^\tess=\{0\}$. To see that it does not depend on the choice of
$b$ in its cohomology class, we suppose that $b=da$, with a
$\Gm$-equivariant~$a$. Then $\om_{P,Q}= \eta_P-\eta_Q$, with
$\eta_P = \left[a(P),v\right]$ also $\Gm$-equivariant, and
\be \label{CPQR} \frac i2 C(P,Q,R) \= \int_P^R \eta_P - \int_P^Q
\eta_P - \int_Q^R \eta_Q \= df(P,Q,R)
\,, \ee
where $f(P,Q) = -\int_P^Q \eta_P$. So
$\bold a:[b]\otimes v \mapsto [C]$ can be extended to give a linear
map $H^1( \Gm;\E_{1-s}) \otimes \Mf_s^0(\Gm)
\rightarrow H^2(\Gm; \CC)$.

We evaluate $C$ on the representative of the fundamental class
in~\eqref{fuclH}:
\be
\begin{aligned}
 C(\fd) &\= \sum_{e\in E} \bigl( C(P_0,P_e,Q_e)-C(\gm_e P_0,P_e,Q_e)
\bigr) \\
&\qquad\hbox{}
+ \sum_{\kp\in \fdcu}\bigl( C(P_0,P_\kp,\pi_\kp^{-1}P_\kp) -
C(P_\kp,\pi_\kp^{-1}P_\kp,\kp)
\bigr)\,,
\end{aligned}
\ee
where we have used the $\Gm$-invariance of~$C$.

Each term in the sum over $e\in E$ contributes:
\[ -2 i \int_{P_e}^{Q_e} \bigl( \om_{P_0,P_e}
- \om_{\gm_eP_0,P_e} \bigr)
\= 2 i \int_e\om_{\gm_eP_0,P_0} \= 2 i \int_e \bigl[
b(\gm_eP_0,P_0),v\bigr]\,. \]
This is equal to the corresponding term in the definition of $\bold d
(b\otimes v)$ in \eqref{bid-def}, provided we use $P_0$ as the base
point in the description of $b$ in the standard model of group
cohomology.

The contribution of the terms for $\kp\in \fdcu$ are also in accordance
with \eqref{bid-def}, as one sees from the following slightly more
complicated computation:
\begin{alignat*}2 \frac1{2 i} &\biggl( C(P_0,P_\kp,\pi_\kp^{-1}P_\kp)
&-&\; C(P_\kp,\pi_\kp^{-1}P_\kp,\kp)
\biggr)
\displaybreak[0]
\\
&\= - \int_{P_\kp}^{\pi_\kp^{-1}P_\kp} \om_{P_0,P_\kp} & +&
\int_{\pi_\kp^{-1}P_\kp}^\kp \om_{P_\kp,\pi_\kp^{-1}P_\kp}
\displaybreak[0]\\
&\= - \int_{P_\kp}^\kp \om_{P_0,P_\kp} + \int_{\pi_\kp^{-1}P_\kp}^\kp
\om_{P_0,P_\kp}&+& \int_{\pi_\kp^{-1}P_\kp}^\kp \bigl(
\om_{P_0,\pi_\kp^{-1}P_\kp} - \om_{P_0,P_\kp}\bigr)
\displaybreak[0]\\
&\= - \int_{P_\kp}^\kp \om_{P_0,P_\kp} &
+&\int_{\pi_\kp^{-1}P_\kp}^\kp \om_{P_0,\pi_\kp^{-1}P_\kp}
\displaybreak[0]\\
&\= \int_{e_\kp}\bigl( - \om_{P_0,P_\kp}+\om_{\pi_\kp P_0,P_\kp}\bigr)&\=&
\int_{e_\kp}\bigl[ b_{\pi_\kp^{-1}},v\bigr]\,.\qed
\end{alignat*}


\sssect{Reformulation with cup product} The map
\[\Mf_s^0(\Gm) \stackrel{\bold q}\lra \Hpar^1(\Gm;\W\om s, \W\sa s)
\lra \Hpar^1(\Gm;\W\sa s)\]
induces a map
\begin{align*}
H^1(\Gm;\E^{-\infty}_{1-s})\;\otimes \;&\Mf^0_s(\Gm)
\stackrel{\mathrm{id}\otimes \bold q}\lra
H^1(\Gm;\E^{-\infty}_{1-s})\otimes \Hpar^1(\Gm;\W\sa s)\\
&\hbox{} \stackrel\cup\lra \Hpar^2(\Gm;\E^{-\infty}_{1-s}\otimes \W\sa
s)\,.
\end{align*}
Since $\E^{-\infty}_{1-s} \cong \V{-\infty}{1-s}$ and $\W\sa s
\cong \V\sa s \subset\V\infty s$, there is a $G$-invariant
$\CC$-bilinear duality
$\langle\dv,\dv\rangle:\E^{-\infty}_{1-s} \times \W\sa s\lra\CC$. On
the subspace $\E^{-\infty}_{1-s}\times \W\om s$ we have the
description in~\eqref{WPdual}:
\[ \langle f,g\rangle \= \bb(s)\, \bt(f,g)
\= \frac{\bb(s)}{\pi i}\, \int_C[f,g]\,,\]
with a suitable contour~$C$. This leads to the following result:
\begin{lem}\label{lem-ce}Fix $f\in \E^{-\infty}_{1-s}$, and define
$g\in \W\sa s$ by $ g(z)\= \int_p [v,q_s(\dv,z)]$, where $v\in
\Mf^0_s(\Gm)$ and $p\in X_1^\tess$. Then
\be\label{du-EW} \langle f,g\rangle\= \bb(s)\, \int_p [v,f]\,.\ee
\end{lem}
\begin{proof}First we suppose that $p\in X_1^{\tess,\hy}$. Then
$\sing(g)
\subset \supp(p)$ which is compact in~$\uhp$, and $g\in \W\om s$. Let
$C$ be a positively oriented closed curve encircling $p$ once. With
\eqref{WPdual} and Theorem~\ref{thm-psC}:
\begin{align*}
\langle f&,g\rangle \= \bb(s)\,\int_C[f,g] \= \frac{\bb(s)}{\pi
i}\int_{z\in C} \biggl[ f(z),\,\int_{z'\in p} \bigl[ v(z'),q_s(z,z')
\bigr]_{z'}\biggr]\\
&\= \frac{ \bb(s)}{\pi i} \int_{z'\in p} \biggl[ v(z'),\, \int_{z\in
C} \bigl[ f(z),q_s(z,z')\bigr]_z\biggr]_{z'} \= \bb(s)\,
\int_p[v,f]\,.
\end{align*}
The legitimicity of the interchange of the order of square brackets
must
(and can!) be checked. Thus, we have obtained \eqref{du-EW} if
$p\in X^{\tess,\hy}_1$.

We still have to consider the case that $p=e(P,\kp)$ with
$P\in X_0^\tess \cap\uhp$ and $\kp\in \cu$. We approximate $p$ by
$p_Q=e(P,Q)$ with $Q\in \uhp$ on $p$. Put, for $Q\in p$ (including
$Q=\kp$):
\begin{align*}
g_Q(z) &\= \int_{e(P,Q)} [v,q_s(\dv,z)]\,,\\
r_Q(\z) &\= \int_{e(P,Q)} [v,R(\z;\dv)^s]\,.
\end{align*}
With \eqref{ex1c} and Proposition~\ref{prop-rqext} we have
$\tpoiss s r_Q= \bb(s)^{-1}\, g_Q$ for all $Q\in p$. If $Q\neq\kp$
then $r_Q\in \V\om s$. Proposition~\ref{prop-Vominf} implies that
$\lim_{Q\rightarrow\kp} r_Q=r_\kp$ in the topology of $\V\infty s$. Let
$\bt\in \V{-\infty}{1-s}$ be such that $f=\poiss{1-s}\bt$.
\begin{align*}
\langle f,g\rangle &\= \langle \bt,\bb(s)\, r_\kp \rangle \= \bb(s)
\lim_{Q\rightarrow\kp} \langle\bt,r_Q\rangle \= \lim_{Q\rightarrow\kp}
\langle f,g_Q\rangle\\
&\= \bb(s) \lim_{Q\rightarrow\kp} \int_{e(P,Q)} [v,f] \= \bb(s)
\int_{e(P,\kp)} [v,f]\,.
\end{align*}
The last equality follows from the fact that $\int_{e(P,\kp)}[f,v]$
converges absolutely.
\end{proof}


\begin{lem}\label{lem-tr3}The following diagram commutes:
\[ \xymatrix{ H^1( \Gm;\E_{1-s}^{-\infty})
\otimes \Mf^0_s(\Gm)
 \ar[d]_{\frac{-\bb(s)}{2i }\,\mathrm{id}\,\otimes\,\bold q }
 \ar[rrd]^{\bold a}
\\
H^1( \Gm;\E_{1-s}^{-\infty}) \otimes \Hpar^1( \Gm;\W\sa s)
\ar[r]_(.56){\cup}
& \Hpar^2(\Gm;\E_{1-s}^{-\infty}\otimes\W\sa s)
\ar[r]_(.63){\text{duality}}
& \Hpar^2( \Gm;\CC)
} \]
\end{lem}
\begin{proof}We start with $b\in Z^1(\Fpt^\tess;\E^{-\infty}_{1-s})$
representing a class in $H^1(\Gm;\E^{-\infty}_{1-s})$, and with
$v\in \Mf_s^0(\Gm)$. The image $\bold q v \in \Hpar^1(\Gm;\W\sa s)$
is represented by $q\in Z^1(\Fpt^\tess;\W\sa s)$ given by
\[ q(y)(z) \= \int_y [ v, q_s(\dv,z) ]
\qquad(y\in X_1^\tess)\,.\]

Let $C_1 \in Z^2(\Fpt^\tess;\CC)$ be the cocycle obtained from the cup
product
\[ C_1(V) \= \Big\langle (b\cup \bold q v)(V)
\Big\rangle
\qquad(V\in X_1^\tess)\,.\]
Our aim is to relate $C_1$ to $C\circ \dt_2$, where $C$ is the cocycle
in \eqref{Ccoc-def} representing $\bold a([b]\otimes\nobreak v)$, and
where $\dt_\pnt: \Fpt^\tess \lra \Fpt^\tess\otimes \Fpt^\tess$ is as
in~\S\ref{ssect-cp}.


It suffices to consider $C_1$ and $C\circ\dt_2$ on the generators in
\eqref{fuclH}, given in the first column of Table~\ref{tab-basxy}.
\begin{table}[ht]
\[
\renewcommand\arraystretch{1.4}
\begin{array}{|cccc|}\hline
\text{basis elt.~$\Dt$}&x\in X_1^{\tess,T}&y\in X_1^\tess&\\ \hline
\Dt(P_0,P_e,Q_e)& e(P_0,P_e)&e&e\in E\\
\Dt(P_0,\gm_e^{-1}Q_e,\gm_e^{-1}P_e)&
-e(P_0,\gm_e^{-1}P_e)& \gm_e^{-1} e(P_e,Q_e)
&e\in E\\
\Dt(P_0,P_\kp,\pi_\kp^{-1}P_\kp) & e(P_0,P_\kp)& f_\kp& \kp\in \fdcu\\
V_\kp& -f_\kp& \pi_\kp^{-1}e_\kp&\kp\in \fdcu
\\\hline
\end{array}
\]
\caption{Basis elements of $F^\tess_2$}\label{tab-basxy}
\end{table}
For each of these generators we have
$(b\cup q)(\Dt) = - b(x)\otimes q(y)$ with $x$ and $y$ as indicated
in Table~\ref{tab-basxy}. Lemma~\ref{lem-ce} shows that
\[ C_1(\Dt) \= - \langle b(x), q(y)\rangle \=
-\bb(s)
\int_y [v,b(x)]\,.\]
With \eqref{om} and \eqref{Ccoc-def} we have
$ C(\Dt) \= -2 i \int_y[b(x),v]$. In view of \eqref{som[]}:
\[ \frac i2 C(\Dt) - \bb(s)^{-1} \, C_1(\Dt)
\= \int_y d\bigl(v\, b(x)
\bigr)\,.\]
A computation on the basis elements of $F_2^\tess$ in
Table~\ref{tab-basxy} shows that
\[ \int_y d\bigl(v\, b(x) \bigr) \=
(dF)(\Dt)\,\]
where $F\in \map(F_1^\tess;\CC)^\Gm$ satisfies
\begin{alignat*}2 F\bigl(e(P_0,P)\bigr)&\=
-v(P)\, b\bigl(e(P_0,P)
\bigr)(P)&\quad&\text{ for all vertices $P$ of $\fd_\hy$}\,,\\
F(e)&\= -v(Q_e)\, b(e)(Q_e)&&\text{ for }e\in E\,,\\
F(f_\kp)&\= -v(P_\kp)\, b(f_\kp)(\pi_\kp^{-1}P_\kp)&&\text{ for }\kp\in
\fdcu\,.
\end{alignat*}
Hence $[C]=-2i \,\bb(s)^{-1} \, [C_1]$. (For $\Dt=V_\kp$ with
$\kp\in \fdcu$ we use that $v(\kp)=0$, and that $F(e_\kp)$ is
unimportant since $F(\pi)\kp^{-1} e_\kp)=F(e_\kp)|\pi_\kp=F(e_\kp)$.)
\end{proof}

\sssect{Combination}We combine the commuting diagrams in the Lemmas
\ref{lem-tr1}, \ref{lem-tr2} and~\ref{lem-tr3}:
\[ \xymatrix{ \Mf_{1-s}^0(\Gm) \otimes \Mf_s^0(\Gm) \ar[d]_{\bold
b\,\otimes\,\mathrm{id}} \ar[rrd]^{\langle\dv,\dv\rangle}
\\
H^1( \Gm;\E_{1-s}^{-\infty}) \otimes \Mf^0_s \ar[d]_{\frac{-\bb(s)}{2i
}\, \mathrm{id}\,\otimes\,\bold q } \ar[rr]^{\bold d} \ar[rrd]^{\bold
a}
&& \CC
\\
H^1( \Gm;\E_{1-s}^{-\infty}) \otimes \Hpar^1( \Gm;\W\sa s)
\ar[r]_(.56){\cup}
& \Hpar^2(\Gm;\E_{1-s}^{-\infty}\otimes\W\sa s)
\ar[r]_(.63){\text{duality}}
& \Hpar^2( \Gm;\CC)
\ar[u]_(.7){\text{eval.~on $(\fd)$}} } \]
Since $\V{-\infty}{1-s} \cong \E^{-\infty}_{1-s}$ and $\V\infty s
\supset \V\sa s \cong \W\sa s$, this completes the proof of
Theorem~\ref{thm-Ppcp}.\qed


\newpage


\skipbetweenchapters
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%    References                                           %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\medskip\bibitem{Sh59} G.~Shimura: {\it Sur les int\'egrales
 attach\'ees aux formes automorphes.} J.\ math.\ Soc.\ Japan~{\bf 11}
 (1959), 291--311. 

\medskip\bibitem{Wat} G.~N.~Watson: {\it A Treatise on the Theory of
 Bessel Functions.} Cambridge, at the Univ.\ Press, 1944.

\medskip\bibitem{Za93}  D.\,Zagier: {\it Periods of modular forms,
 traces of Hecke operators, and multiple zeta values.} In {\it
Hokei-keishiki to L-kansuu no kenkyuu} (= Research on Automorphic
Forms and L-Functions), RIMS Kokyuroku 843 (1993), 162-170. 

\medskip\bibitem{Za2} D.\,Zagier: {\it New points of view on the
Selberg zeta function.} Proceedings of the Japanese-German Seminar
"Explicit Structures of Modular Forms and Zeta Functions", Ryushi-do
(2002), 1-10. 

\medskip\bibitem{Za10} D.~Zagier: {\it Quantum modular forms.}
In {\it Quanta of Maths: Conference in Honor of Alain
Connes}, Clay Mathematics Proceedings 11, AMS and Clay Mathematics
Institute 2010, 659-675. 

\end{thebibliography}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%    Index                                                %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\renewcommand\indlbl[2]{\item #1\quad\ #2}
\newcommand\ir[1]{\pageref{ind-#1}}
\begin{theindex}
\indlbl{analytic vectors in principal series}{\ir{anv}}
\indlbl{antisymmetrization}{\pageref{antisym}}
\indlbl{attracting fixed point}{\ir{rfp}}
\indlbl{augmentation}{\ir{augm}}
\indlbl{augmentation ideal}{\ir{gr}}
\indlbl{average}{\ir{av}, \pageref{avGdef}}
\indexspace
\indlbl{base point}{\ir{bp}}
\indlbl{bilinear Petersson product}{\ir{bPp}}
\indlbl{boundary germ}{\ir{bg}}
\indlbl{boundary jet}{\ir{Wpdef}}
\indlbl{boundary map}{\ir{pibd}}
\indlbl{boundary of hyperbolic plane}{\pageref{uhp-disk}}
\indexspace
\indlbl{cap product}{\ir{cp}}
\indlbl{$1$-chain}{\ir{1ch}}
\indlbl{chain complex on $\H$}{\ir{fhyp}}
\indlbl{circle model of principal series}{\ir{cm}}
\indlbl{coboundary}{\ir{cb}}
\indlbl{cochain}{\ir{cch}}
\indlbl{cocompact}{\ir{cc}}
\indlbl{cocycle relation}{\pageref{c-co}}
\indlbl{cofinite}{\ir{cof}}
\indlbl{cohomology group}{\ir{cg}, \ir{fco}}
\indlbl{coinvariants}{\ir{coinv}}
\indlbl{commutator subgroup}{\ir{Gcom}}
\indlbl{cup product}{\ir{cup}}
\indlbl{cup product for parabolic cohomology}{\ir{cuppar}}
\indlbl{cusp}{\ir{cusp}}
\indlbl{cusp form}{\ir{cf}}
\indlbl{$1$-cycle}{\ir{1ch}}
\indlbl{cyclic interval}{\ir{cint}}
\indexspace
\indlbl{Dirichlet fundamental domain}{\ir{Di}, \ir{D}}
\indlbl{disk model of hyperbolic plane}{\ir{disk}}
\indlbl{distribution vectors in principal series}{\ir{di}}
\indlbl{duality}{\ir{du}, \pageref{WEdual}}
\indexspace
\indlbl{edge}{\ir{pol}}
\indlbl{$\ld_s$-eigenfunction}{\ir{ldsef}}
\indlbl{Eisenstein series}{\ir{Eis}}
\indlbl{excised neighborhood}{\pageref{def-wn}}
\indlbl{extended upper half-plane}{\ir{uhpa}}
\indexspace
\indlbl{Fourier expansion}{\pageref{ssect-Fe}}
\indlbl{free-space resolvent kernel}{\ir{frk}}
\indlbl{fundamental class}{\ir{fc}, \pageref{fc-c}, \pageref{fuclH}}
\indlbl{fundamental domain}{\ir{fd},
\ir{fdc}}
\indexspace
\indlbl{geodesic}{\ir{geo}}
\indlbl{Green's form}{\ir{Gsfi}, \ir{Gf}}
\indlbl{group ring}{\ir{gr}}
\indexspace
\indlbl{homogeneous cocycle}{\ir{hc}}
\indlbl{homology group}{\ir{cg}}
\indlbl{horocycle}{\ir{horocycle}}
\indlbl{horocyclic disk}{\ir{horocycle}}
\indlbl{hyperbolic distance}{\pageref{uhp-disk}}
\indlbl{hyperbolic element}{\pageref{ssect-avhyp}}
\indlbl{hyperbolic fixed point}{\ir{hyp}}
\indlbl{hyperbolic plane}{\ir{hp}}
\indlbl{hyperfunction}{\ir{hfs}}
\indexspace
\indlbl{inhomogeneous cocycle}{\ir{ic}}
\indlbl{intertwining operator}{\ir{I}}
\indlbl{invariant eigenfunction}{\ir{iei},
\ir{ie}}
\indlbl{invariants}{\ir{inv}}
\indlbl{isometry group}{\pageref{uhp-disk}}
\indexspace
\indlbl{Laplace operator}{\pageref{uhp-disk}}
\indlbl{line model of principal series}{\ir{lm}}
\indlbl{locally defined module}{\pageref{def-locdef}}
\indexspace
\indlbl{Maass cusp form}{\ir{Mcfi},
\ir{Mcfii}, \ir{cf}}
\indlbl{Maass form}{\ir{Mwfi}, \ir{Mfii},
\ir{Maf}}
\indlbl{Maass-Selberg relation}{\ir{MS}}
\indlbl{mixed parabolic cocycle}{\ir{defpcG}}
\indlbl{mixed parabolic cohomology group}{\ir{mpcgi}, \ir{defpcG},
\ir{mpcg}}
\indlbl{modular group}{\ir{mgi0}, \ir{mgi},
\ir{mg}, \pageref{sect-pf}}
\indexspace
\indlbl{one-sided average}{\ir{osa}}
\indexspace
\indlbl{parabolic cocycle}{\ir{pc}}
\indlbl{parabolic cohomology group}{\ir{pcgi}, \ir{pc},
\pageref{pcdef}}
\indlbl{parabolic element}{\ir{T}}
\indlbl{parabolic fixed point}{\ir{pfp}}
\indlbl{parabolic resolution}{\ir{pres}}
\indlbl{period function}{\ir{pfi}, \pageref{gc0}, \pageref{thm-pfsm}}
\indlbl{periodlike function}{\ir{plf}}
\indlbl{Petersson product}{\ir{Pp}}
\indlbl{Poincar\'e series}{\ir{Poinc}}
\indlbl{point-pair invariant}{\ir{ppi}}
\indlbl{Poisson transformation}{\ir{Po}}
\indlbl{polygon}{\ir{pol}}
\indlbl{polynomial growth}{\ir{pg},
\ir{pogr}}
\indlbl{principal series}{\ir{spr}}
\indlbl{projective line}{\ir{prR}}
\indlbl{projective model of principal series}{\ir{pm}}
\indexspace
\indlbl{quantum Maass form}{\ir{qmf}}
\indlbl{quick decay}{\ir{qd}}
\indexspace
\indlbl{reconstruction}{\pageref{ssect-recon}}
\indlbl{refinement}{\ir{ref}}
%\indlbl{relative cohomology group}{\ir{rc}}
\indlbl{repelling fixed point}{\ir{rfp}}
\indlbl{restriction of boundary germs}{\ir{resbd}}
\indlbl{right module}{\ir{rm}}
\indexspace
\indlbl{second order eigenfunction}{\ir{soef}}
\indlbl{semi-analytic vector in principal series}{\ir{savi}, \ir{sav}}
\indlbl{seminorms on $\V p s$ and $\V\infty s$}{\ir{snj}}
\indlbl{sheaf cohomology}{\ir{shcoh}}
\indlbl{singularity}{\ir{Vbs}, \ir{sing1}, \pageref{singdef}}
\indlbl{smooth semi-analytic vectors in principal series}{\ir{smsav}}
\indlbl{smooth vectors in principal series}{\ir{Vsoi}}
\indlbl{spectral parameter}{\ir{spi},
\ir{spr}}
\indlbl{standard fundamental domain}{\ir{stfd}}
\indlbl{standard model of group cohomology}{\ir{sm}}
\indlbl{standard resolution}{\ir{Fgc}}
\indlbl{support of chain}{\ir{su}}
\indexspace
\indlbl{tesselation}{\ir{te}, \ir{t2}}
\indlbl{three term equation}{\ir{ttei}, \pageref{3TC}}
\indlbl{transverse Poisson transformation}{\ir{tP}}
\indlbl{triangulation}{\ir{tr}}
\indlbl{type of tesselation}{\pageref{ssect-cc-ex}}
\indexspace
\indlbl{upper half-plane model of hyperbolic plane}{\ir{uhp}}
\indexspace
\indlbl{vertex}{\ir{vert}}
\indlbl{volume element}{\pageref{uhp-disk}}
\indexspace
\indlbl{wave form}{\ir{Mwfi}}
\indlbl{winding number}{\ir{wn}}
\end{theindex}

\section*{Notations}
\renewcommand\indlbl[2]{\item $#1$\quad\ #2}
\begin{multicols}{3}
\raggedright
\begin{trivlist}\footnotesize
\indlbl{A\;\subset\; G}{\ir{Adef}}
\indlbl{A\;\subset\;\Gm}{\ir{A}, \ir{Ac}}
\indlbl{A_\pnt}{\pageref{antisym}}
\indlbl{A_n,\, A_\al}{\pageref{Fourexp-par}}
\indlbl{\av g,\; \av g^+,\; \av g^-}{\pageref{av-def},
\pageref{reg-av}}
\indlbl{\av \Gm}{\pageref{avGdef}}
\indlbl{\av T^\uparrow(\dv)\,,\;\av T^\downarrow(\dv)}{\ir{avTud}}
\indlbl{a_\pnt}{\pageref{a-def}}
\indexspace
%\indlbl{\Bcal^p_s}{\irr{Bps}}
\indlbl{B}{\pageref{Efd}}
\indlbl{B^i( \Gm;V), B^i(\Fpt;V)}{\pageref{H1def}, \ir{fcob}}
\indlbl{B^i(\Fpt;V,W)}{\pageref{mpcgd}}
\indlbl{B_n,\, B_\al}{\pageref{Fourexp-par}}
\indlbl{\bsing}{\ir{Vbs}, \ir{bsing}}
\indlbl{\bold b}{\ir{bbd}}
\indlbl{\bb(s)}{\pageref{btd}}
\indexspace
\indlbl{\cu}{\ir{cu}}
\indlbl{\CC'\;=\; \CC\sm(-\infty,0]}{\ir{plf}}
\indlbl{C^\om}{\ir{Com}}
\indlbl{C^{-\om}}{\ir{hfs}}
\indlbl{C_+,\; C_-}{\ir{C+C-}}
\indlbl{C^i(\Fpt;V)}{\ir{fcoc}}
\indlbl{C^i(\Fpt;V,W)}{\pageref{mpcdef}}
\indlbl{c(s)}{\pageref{splitting}}
\indexspace
\indlbl{\disk}{\ir{disk}}
\indlbl{\Dcoh}{\ir{typeD}, \ir{Dcohc}}
\indlbl{D_a(\kp)}{\ir{horocycle}}
\indlbl{d^i}{\ir{dic}, \pageref{mpcdef}}
\indlbl{\dist(\dv,\dv)}{\pageref{uhp-disk}}
\indexspace
\indlbl{\E_s}{\ir{E}}
\indlbl{\E_s(\dv)}{\ir{Esh}}
\indlbl{\E^{-\infty}_s,\ \E^\infty_s,\; \E^\om_s}{\ir{E*}}
\indlbl{\E_s^\uparrow\,,\; \E_s^\downarrow}{\pageref{Eupdown}}
\indlbl{\E_s'\,,\;
(\E_s')^{-\infty}}{\pageref{E'def}}
\indlbl{E}{\ir{Ee}, \ir{Ec}}
\indlbl{E_s^\kp}{\ir{Eis}}
\indlbl{\eph_{s,n},\; \eph_{s,n}^\prj,\;
\eph_{s,n}^\ci}{\pageref{phisndef}}
\indlbl{e_{P,Q}\;=\; e(P,Q)}{\ir{ePQ}}
\indlbl{e_\kp \;=\; e(P_\kp,\kp)}{\ir{efkap}}
\indexspace
\indlbl{\F_s}{\pageref{F-def}}
\indlbl{\sh_{V,W}}{\pageref{sheafdef}}
\indlbl{\Fcoh}{\ir{typeF}, \ir{Fcohc}}
\indlbl{\fd}{\ir{fd}, \ir{fdc}}
\indlbl{\fdcu}{\ir{fdcu}}
\indlbl{\fd_\hy}{\ir{fdgy}}
\indlbl{\Fpt^\gcoh}{\ir{Fgc}}
\indlbl{\Fpt^\hcoh}{\ir{fhyp}, \ir{hcohc}}
\indlbl{\Fpt^{\hcoh-}}{\ir{Fhyp-}}
\indlbl{\Fpt^{\hcoh,\hy}}{\ir{hcohc}}
\indlbl{\Fpt^\tess}{\ir{F-Tcal},
\ir{F-Tcalc}}
\indlbl{\Fpt^{\tess,\hy}}{\ir{Ffcc}}
\indlbl{\pf_s(\dv)}{\ir{plf}}
\indlbl{\pf_s(\dv)_\om\,,\; \pf_s(\dv)_\infty}{\ir{pfss}}
\indlbl{\pf_s(\dv)^0_\pnt}{\pageref{gc0}}
\indlbl{\pf(\RR_+)_\om^1}{\ir{gc+}}
\indlbl{f^\ci\,,\; f^\prj\,,\; \tilde f}{\ir{Wom}, \ir{Gd}}
\indlbl{f_\kp \;=\; e(P_\kp,\pi_\kp^{-1}P_\kp)}{\ir{efkap}}
\indlbl{f_{s,\al}^R,\; f_{s,\al}^L}{\pageref{frt}}
\indexspace
\indlbl{\G^\om_s, \G^p_s}{\ir{Gomi}, \ir{Gd}}
\indlbl{\G^p_s(I)}{\ir{GNI}}
\indlbl{\G^\fsh_s}{\pageref{Gfsh}}
\indlbl{\G^\wdgwdg_s\,,\; \G^\wdg_s}{\pageref{Gwdg-def}}
\indlbl{G}{\ir{G}}
\indlbl{g_\kp}{\ir{gkap}}
\indexspace
\indlbl{\H}{\ir{HH}}
\indlbl{\HHH,\; \HHH_s}{\ir{HHH}}
\indlbl{\uhp}{\ir{uhp}}
\indlbl{\uhp^\ast}{\ir{uhpsa}}
\indlbl{\uhp_a}{\ir{uhpa}}
\indlbl{H^\hpnt( \Gm;V),\; H_\pnt(\Gm;V)}{\pageref{chd}}
\indlbl{H^1(\Dt;V,W)}{\pageref{mcd}}
\indlbl{\Hpar^i(\Gm;V)}{\ir{pc}, \pageref{pcdef}}
\indlbl{\Hpar^i(\Gm;V,W)}{\ir{defpcG}, \pageref{mpcgd}}
\indlbl{H_a(\kp)}{\ir{horocycle}}
\indexspace
\indlbl{\I s}{\pageref{Edecomp}}
\indlbl{\I s^\downarrow}{\ir{Id}}
\indlbl{I_s}{\ir{I}}
\indlbl{I_\mu(\dv)}{\ir{IK-Bessel}}
\indlbl{i_{s,\al}}{\pageref{ikdef}}
\indexspace
\indlbl{\bold j}{\pageref{jj}}
\indexspace
%\indlbl{\Kcal_A}{\pagereff{KA-seq}}
\indlbl{\K_s\,,\; \K^0_s}{\pageref{Edecomp}}
\indlbl{\K_s^\uparrow\,,\;
(\K^0_s)^\uparrow}{\ir{Ku0}}
\indlbl{K\;\subset\; G}{\ir{Kdef}}
\indlbl{K_\mu(\dv)}{\ir{IK-Bessel}}
%\indlbl{k(\dv)}{\pagereff{tab-opsg}}
\indlbl{k_{s,\al}}{\pageref{ikdef}}
\indexspace
\indlbl{\ell_{s,0}}{\ir{ell-h}}
\indexspace
\indlbl{\Mcoh}{\ir{typeM}, \ir{Mcohc}}
\indlbl{\Mcoha}{\ir{typeMa}, \ir{Mcohac}}
\indlbl{\Mf_s(\Gm)}{\ir{Mf}}
\indlbl{\Mf_s^0(\Gm)}{\ir{Moi}, \ir{cf}}
\indlbl{\Mf^1_s(\Gm)}{\ir{Mf1}}
\indlbl{\map(X,V)}{\ir{map}}
\indlbl{\bold m}{\ir{mm}}
\indlbl{m_C}{\ir{wn}}
\indexspace
\indlbl{\N^\om, \N^p_s}{\ir{Nomi}\, \ir{Nd}}
\indlbl{\N^p_s(I)}{\ir{GNI}}
\indlbl{\N^\wdg}{\ir{Nwdg}}
\indlbl{\NN}{\ir{NN}}
\indlbl{N\;\subset\; G}{\ir{Ndef}}
\indlbl{N_R(\dv)}{\ir{NR}, \pageref{NRcusp}}
\indexspace
\indlbl{\hol(\dv)}{\ir{hol}}
\indexspace
%\indlbl{\P^\fs_s}{\pagereff{Pomastdef}}
%\indlbl{\P^\om_s[\x_1,\ldots,\x_n]}{\irr{Pom[]}}
%\indlbl{\P^\cptwdg}{\irr{Siegel}}
\indlbl{\proj\RR \;=\; \p\uhp}{\ir{prR}}
%\indlbl{P^n_{-s}}{\ir{Lf}}
\indlbl{P_{\!s,n}}{\pageref{psn-def}}
\indlbl{P_\kp}{\ir{Pkap}}
\indlbl{P_e}{\ir{PQe}}
\indlbl{P_s}{\ir{Poinc}}
\indlbl{\poiss s}{\ir{Potr}}
\indlbl{\tpoiss s}{\pageref{Wdef}, \pageref{rTW}, \ir{rWsmp},
\ir{tp-wdg}}
\indlbl{\bold p}{\ir{pqr}}
\indlbl{p_s}{\pageref{pqs-def}}
\indlbl{p_{P,Q}}{\pageref{pqr-geom}}
\indlbl{(p_\gm)}{\pageref{pdef}}
\indexspace
\indlbl{\qMf_s(\Gmod)}{\pageref{qmfnot}}
\indlbl{\Q_s}{\ir{Qsdef}}
\indlbl{Q_e}{\ir{PQe}}
\indlbl{Q_{s,n}}{\pageref{qsn-def}}
\indlbl{\bold q}{\ir{pqr}, \pageref{rqT}}
\indlbl{q_s}{\pageref{pqs-def}}
\indlbl{(q_\gm)}{\pageref{qdef}}
\indlbl{q_{P,Q}}{\pageref{pqr-geom}}
\indexspace
\indlbl{\R_s}{\ir{Rsdef}}
\indlbl{R(\dv;\dv)^s}{\pageref{R-def}}
\indlbl{R^\prj(\dv;\dv)^s,\;
R^\ci(\dv;\dv)^s}{\pageref{R-othermodels}}
\indlbl{\bold r}{\ir{pqr}, \pageref{lem-uupbt}, \pageref{rqT}}
\indlbl{(r_\gm)}{\pageref{rdef}}
\indlbl{r_{P,Q}}{\pageref{pqr-geom}}
\indexspace
\indlbl{\Scal_s\,,\; \Scal_{s,\x}}{\ir{singq}}
\indlbl{\Scal^W_s}{\pageref{def-locdef}}
\indlbl{\Scal^W_{s,\x}}{\ir{SlocW}}
\indlbl{\uc\;=\; \p\,\disk}{\ir{uc}}
\indlbl{S\;=\; \matr0{-1}10}{\ir{Sgn1}}
\indlbl{\supp}{\ir{su}}
\indlbl{\sing}{\pageref{singdef}}
\indlbl{s}{\ir{s}, \ir{spr}}
\indexspace
\indlbl{T\;=\; \matc 1101}{\ir{T}, \ir{Teta},
\ir{Tgn1}}
\indlbl{T'\;=\; \matc1011}{\ir{Ta}}
\indexspace
\indlbl{u_D,\, u_U,\, U_L,\, u_R}{\ir{uDULR}}
\indlbl{u_\ps,\; u_{[\ps]}}{\ir{u-ps},
\ir{upa}, \pageref{upsdef-cp}}
\indexspace
\indlbl{\V{}s}{\ir{Vs}}
\indlbl{\V\infty s}{\ir{Vsoi}}
\indlbl{\V\om s}{\ir{Vomi}, \ir{anv}}
\indlbl{\V p s}{\ir{Vp}}
\indlbl{\V{-\infty}s}{\ir{V-ii}, \ir{di}}
\indlbl{\V{-\om}s}{\ir{V-om}}
\indlbl{\V\om s(I)}{\pageref{Vsheaf}}
\indlbl{\V\om s[F],\; \V{\om,\mathrm{cond}}s}{\pageref{V[]}}
\indlbl{\V\fs s}{\pageref{Vfs}}
\indlbl{\V\sa s}{\ir{Vsai}, \ir{Vsa}}
\indlbl{\V\smp s}{\pageref{V(1)def}}
\indlbl{\V\wdg s}{\ir{vssgi}, \pageref{wdgdef}}
\indlbl{\V{\om^0}s, \;\V\sacp s}{\ir{Vom0}}
\indlbl{ \V\smpcp s,\;\V\wdgcp s}{\ir{Vom0}}
\indlbl{V^g \;=\; V^{\gen g}}{\ir{inv}}
\indlbl{V_\Gm}{\ir{coinv}}
\indlbl{V_\kp}{\ir{Vkap}}
%\indlbl{v_\ps}{\irr{vpsi}}
%\indlbl{v_{\tilde\ps}[\x,\eta]}{\irr{vtpsd}}
\indexspace
\indlbl{\W p s, \W p s(I)}{\ir{Wpdef}}
\indlbl{\W\om s, \W\om s(I)}{\ir{Womi},
\ir{Wom}}
\indlbl{\W\om s[F],\; \W\fs s}{\pageref{Wfs-def}}
\indlbl{\W\sa s}{\ir{Wsa}}
\indlbl{\W\smp s}{\pageref{V(1)def}}
\indlbl{\W\wdg s}{\ir{Wwdg}}
\indlbl{w\;\text{ coord.~on $\disk$}}{\pageref{uhp-disk}}
\indexspace
\indlbl{X_\pnt\,,\; X_\pnt^\tess}{\ir{pol}, \pageref{XXY}}
\indlbl{X_\pnt^\hy}{\pageref{XXY}}
\indlbl{X_0^H}{\ir{Xh0}}
\indlbl{\xg}{\ir{XX}}
\indlbl{x_\kp}{\ir{xkap}}
\indexspace
\indlbl{\hy}{\ir{hy}}
\indlbl{\yg,\;\yg_{\!\hy}}{\ir{YY}}
\indexspace
\indlbl{\ZZ[\Gm]}{\ir{gr}}
\indlbl{Z^i( \Gm;V), Z^i(\Fpt;V)}{\pageref{H1def}, \ir{fcoz}}
\indlbl{\Zpar^i(\Gm;V,W)}{\ir{defpcG}}
\indlbl{Z^i(\Fpt;V,W)}{\pageref{mpcgd}}
\indlbl{z\;\text{ coord.~on $\H$ and $\uhp$}}{\ir{zco}}
\indexspace
\indlbl{\al(\eta)}{\ir{rfp}}
\indlbl{\al(u)}{\pageref{aldef}}
\indlbl{\al_s^\om}{\ir{al-cp}, \ir{al-cp1}}
\indlbl{\al_s^p}{\ir{alp}}
\indexspace
\indlbl{\bt(u)}{\pageref{btdef}, \pageref{lem-uupbt}}
\indlbl{\bt^R(u)\,,\;\bt^L(u)}{\pageref{btlr}}
\indexspace
\indlbl{\Gm}{\ir{gam}, \ir{gam1}}
\indlbl{\Gm_\kp}{\ir{itrsg}}
\indlbl{\Gmod\;=\; \PSL_2(\ZZ)}{\ir{mgi0},
\ir{mg}}
\indlbl{\Gcom\;=\; [\Gmod,\Gmod]}{\ir{Gcom}}
\indlbl{\gm_e}{\ir{Ec}}
\indexspace
\indlbl{\Dt}{\pageref{uhp-disk}}
\indlbl{\Dt\text{ in Chap~\ref{ch-III}}}{\ir{Teta}}
\indlbl{\Dt(\dv,\dv,\dv)}{\ir{tr}}
\indexspace
\indlbl{\e}{\ir{augm}, \ir{fhyp},
\ir{augmcp}}
\indexspace
\indlbl{\eta = \matc {\sqrt t}00{1/\sqrt t}}{\pageref{ssect-avhyp},
\ir{Teta}}
\indexspace
\indlbl{\ld_s}{\ir{lds}}
\indexspace
\indlbl{d\mu}{\pageref{uhp-disk}}
\indexspace
\indlbl{\pi_\kp}{\ir{pikap}}
%\indlbl{\pi_A}{\pagereff{KA-seq}}
\indexspace
\indlbl{\rho(\dv,\dv)}{\pageref{uhp-disk}}
\indlbl{\rho_s}{\ir{rho-def}, \ir{r2}, \pageref{rTW}, \ir{rWsmp}}
\indexspace
\indlbl{\om(\eta)}{\ir{afp}}
\indexspace
\indlbl{g\mapsto v\,|\,g}{\ir{ra}}
\indlbl{\ph \mapsto \ph|_{2s}\,g}{\pageref{tr-lm}}
\indlbl{\ph \mapsto \ph|_{2s}^\prj \,g}{\pageref{projac}}
\indlbl{\ph \mapsto \ph|^\ci_{2s}\, g}{\pageref{trfcimo}}
\indlbl{\ph \mapsto \ph^\prj}{\pageref{phprjd}}
\indlbl{\ph\mapsto \ph^\ci}{\pageref{phci}}
\indlbl{\tilde g}{\ir{gtld}}
\indlbl{\cup \text{ \ (cup product)}}{\ir{cup}, \ir{cuppar}}
\indlbl{(b,a)_c}{\ir{cint}}
\indlbl{(\dv,\dv)}{\ir{Pp}}
\indlbl{\gen g}{\ir{gen}}
\indlbl{\langle\dv\rangle}{\ir{lin}}
\indlbl{\langle \dv,\dv\rangle}{\pageref{dual-ci}, \pageref{WEdual},
\ir{bPp}}
\indlbl{\matc \ast\ast\ast\ast}{\ir{mat}}
\indlbl{[\dv,\dv],\; \{\dv,\dv\}}{\pageref{[]def}}
\indlbl{[\Gm\backslash\H]}{\ir{fc}}
\indlbl{\|\ph\|_j}{\ir{snj}}
\indlbl{\p_i}{\ir{pibd}, \ir{fhyp}, \ir{cpd}}
\indlbl{\mathring\fd,\;\mathring e \ \text{(interior)}}{\ir{int},
\ir{inte}}
\end{trivlist}
\end{multicols}








\end{document}


