
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{An overview of Siegel-Veech constants}
\label{sec:SV}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let $(X,\omega)$ be a flat surface, consisting of a Riemann surface~$X$ 
and an Abelian differential~$\omega$ on~$X$. Siegel-Veech constants
measure the asymptotic growth rate of the number of saddle connections
(abbreviated s.c.) or cylinders with bounded length (of the waist curve)
in~$(X,\omega)$. There are many variants that we now introduce and compare.


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\subsection{Saddle connection and area Siegel-Veech constants}
%%%%%%%%%%%%%%%%%%%%%

For each pair of zeros~$(z_1,z_2)$ of $\omega$ we let
\be \label{eq:Aphy}
A^\phy_{1 \lra 2}(T) \= |\{\gamma \subset X \,\, \text{a saddle connection
joining $z_1$ and $z_2$},\, \Bigl|\int_\gamma \omega \Bigr| \leq T\}|
\ee
be the counting function. The upper index emphasizes that we count
all physically distinct saddle connections. It should be distinguished
from the version
\be \label{eq:Aho}
A^\ho_{1 \lra 2}(T) \= |\{\gamma \subset X \,\, \text{a homology
  class\ of s.c.\ joining $z_1$ and $z_2$},\, \Bigl|\int_\gamma \omega \Bigr|
\leq T\}|\,,
\ee
where a collection of homologous saddle connections just counts for one.
Quadratic upper and lower bounds for such counting
functions were established by Masur (\cite{masur90}). Fundamental works of
Veech (\cite{veech98}) and Eskin-Masur (\cite{eskinmasur}) showed that for
almost every flat surface $(X,\omega)$ in the sense of the Masur-Veech measure
(see \cite{masur82} and \cite{veech82}) there is a quadratic asymptotic, i.e. that
\be \label{eq:defscSV}
A^\phy_{1 \lra 2}(T) \, \sim \, c^\phy_{1 \lra 2}(X,\omega) \,\pi T^2\,,
\qquad A^\ho_{1 \lra 2}(T) \, \sim \, c^\ho_{1 \lra 2}(X,\omega) \,\pi T^2\,.
\ee
The constants $c^\phy_{1 \lra 2}(X,\omega)$ and $c^\ho_{1 \lra 2}(X,\omega)$
are the first type of Siegel-Veech constants we study here, called the
{\em saddle connection Siegel-Veech constants}. The difference between
these two Siegel-Veech constants becomes negligible as the genus of~$X$
tends to infinity, which follows from the results of Aggarwal and
Zorich (see~\cite[Remark 1.1]{Agg2}).
\par
\medskip
The second type of Siegel-Veech constants counts homotopy classes of closed
geodesics, or equivalently flat cylinders. Again, there are two variants, the
naive count and the count where each cylinder is weighted by
its relative area. As above, the most important counting function
with good properties (see e.g.\ \cite{cmz}) and connection to Lyapunov
exponents (\cite{ekz}) is the second variant. For the precise definition
we consider
\be \label{eq:Narea}
A_{\cyl}(T) \= \sum_{Z \subset X \text{cylinder} \atop w(Z) \leq T} \, 1\,,
\qquad
A_{\area}(T) \= \sum_{Z \subset X \text{cylinder} \atop w(Z) \leq T}
\frac{\area(Z)}{\area(X)}\,,
\ee
where $w(Z)$ denotes the width of $Z$, i.e. the length of its core curve. We then 
define the {\em cylinder Siegel-Veech constant} and the
{\em area Siegel-Veech constant} by the asymptotic equalities
\be \label{eq:SVarea}
A_{\cyl}(T) \, \sim \, c_{\cyl}(X,\omega){\pi T^2} \,, \qquad 
A_{\area}(T) \, \sim \, c_{\area}(X,\omega){\pi T^2} \,.
\ee
\par
There is a natural action of $\GL_2(\RR)$ on the moduli space of flat
surfaces $\omoduli[g]$ and the orbit closures are nice submanifolds,
in fact linear in period coordinates by the fundamental work of Eskin, Mirzakhani and Mohammadi (\cite{esmi} and \cite{esmimo}). We
refer to them as {\em affine invariant manifolds}, using typically the
letter~$\cMM$. The intersection with the hypersurface of area one flat surfaces
(denoted by the same symbol~$\cMM$) comes with a finite $\SL_2(\RR)$-invariant
ergodic measure $\nu_\cMM$ with support~$\cMM$. This measure is unique up 
to scale and for affine invariant manifolds defined over~$\QQ$ there 
are natural choices of the scaling.
\par
It follows from the Siegel-Veech axioms 
(see \cite{eskinmasur}) that Siegel-Veech constants for almost all
flat surfaces $(X,\omega)$ in an $\SL_2({\RR})$-orbit closure~$\cMM$ agree.
We call these surfaces {\em generic} (for $\cMM$) and write e.g.\
$ c_{1 \lra 2}^\star(\cMM) = c_{1 \lra 2}^\star(X,\omega)$ for $(X,\omega)$ generic.
\par
The relevant orbit closures in this paper are the connected components of
the strata of Abelian differentials and certain Hurwitz spaces inside the strata.
We usually abbreviate by $c_{1 \lra 2}^\star(\mu) =
c_{1 \lra 2}^\star(\omoduli[g,n](\mu))$ the Siegel-Veech constants for the strata with signature $\mu$.

%%%%%%%%%%%%%%%%%%%%%
\subsection{Configurations and the principal boundary}
\label{ssec:configurations}
%%%%%%%%%%%%%%%%%%%%%

One of the main insights of \cite{emz} is that Siegel-Veech constants
can be computed separately according to topological types, called
configurations. We formalize their notion of configurations briefly so that
it also applies to Hurwitz spaces, and in fact to all $\SL_2(\RR)$-orbit
closures~$\cMM$ provided with the generalization of the Masur-Veech
measure~$\nu_\cMM$. The concept of configurations will be used for showing
the equivalence between Theorem~\ref{intro:SVproduct} and
Theorem~\ref{intro:VolRec} in Section~\ref{sec:VRandSC}.
\par
Let $(X,z_1,\ldots,z_n)$ be a pointed topological surface. A 
{\em configuration $\cCC$ of saddle connections joining~$z_1$ and~$z_2$}
for~$\cMM$ is a set of simple non-intersecting arcs from~$z_1$ to~$z_2$
up to homotopy preserving the cyclic ordering of the arcs both at~$z_1$
and~$z_2$. The last condition implies that the tubular neighborhood of
the configuration is a well-defined subsurface of~$X$, in fact a
{\em ribbon graph~$R(\cCC)$} associated with the configuration.
(Here the subsurfaces of $X$ after removing a ribbon graph are
allowed to have positive genus.) The number of arcs in the configuration
is called the {\em multiplicity} of the configuration.
\par
We say that a subset of the saddle connections joining~$z_1$
and~$z_2$ on a flat surface $(X,\omega)$ {\em belongs} to the
configuration $\cCC$, if this subset is homotopic to~$\cCC$.
Each configuration gives rise to a counting function $A_{1 \lra 2}^\star
(T,\cCC)$ for sets of saddle connections belonging to the
configuration and having individual length $\leq T$.
From the counting function we deduce the
corresponding Siegel-Veech constant $c_{1 \lra 2}^\star(\cMM,\cCC)$, where
$\star \in \{\phy,\ho\}$ respectively. A configuration~$\cCC$ is {\em relevant}
if $c^\star_{1 \lra 2}(\cMM,\cCC) >0$.
\par
A {\em full set of saddle connection  configurations} for an affine invariant
manifold $\cMM$ is a finite set of saddle connection configurations $\cCC_i$,
with $i\in I$ such that the contributions of the configurations $\cCC_i$ sum up to
the full Siegel-Veech constant, i.e.\ such that
\be \label{eq:SV12sum}
\sum_{i \in I} c^\star_{1 \lra 2}(\cMM,\cCC_i) \= c^\star_{1 \lra 2}(\cMM)
\ee
for $\star \in \{\phy,\ho\}$ respectively.
\par
Note that \cite[Section~3.2]{emz} in their definition of configurations
made a further subdivision of the notion by adding metric data, i.e.\ specifying angles
between saddle connections. In that context, Eskin, Masur and Zorich determined a full
set of saddle connection configurations for the strata and used
the Siegel-Veech transform to connect the computation to volume
computations. The following statement summarizes 
Proposition~3.3, Corollary~7.2 and Lemma~8.1 of \cite{emz}.
\par
\begin{Prop} \label{prop:emzsummary}
For any stratum~$\omoduli[g,n](\mu)$ a full set of saddle connection
configurations is the set of collections of pairwise homologous simple
disjoint arcs joining~$z_1$ and $z_2$ (up to homotopy).
\end{Prop}
\par
In this proposition, several configurations are irrelevant, for example
those with a connected component of genus zero after removing the saddle connections in the configuration. 
\par
The general strategy to compute Siegel-Veech constants is the following
relation to volumes, where the submanifold $\cMM$ is in
a stratum with labeled zeros.
\par
\begin{Prop} \label{prop:SVviaboundary}
  The saddle connection Siegel-Veech constants of an
affine invariant manifold~$\cMM$ can be computed as 
\be \label{eq:SVviaboundary}
c_{1 \lra 2}^\star (\cMM) \= \lim_{\ve \to 0} \frac1{\pi\ve^2} \sum_\cCC
m^\star(\cCC)\,
\frac{\nu_{\cMM}(\cMM^\ve(\cCC))}
{\nu_{\cMM}(\cMM)}\,,
\ee
where the sum runs over the full set of saddle connection configurations
and where $m^\ho(\cCC) = 1$ for all $\cCC$ while $m^\phy(\cCC)$ is equal
to the number of arcs in~$\cCC$.
\end{Prop}
\par
\begin{proof}
This is a direct consequence of the Siegel-Veech transform applied
to the characteristic function of a disc of radius~$\ve$, see
\cite[Lemma~7.3]{emz} together with the Eskin-Masur bound on the
number of short saddle connections (Theorem on p.~84 of \cite{emz}).
\end{proof}
\par
We conclude with remarks on Siegel-Veech constants for general affine 
invariant manifolds to put the digression on Hurwitz spaces 
(Section~\ref{sec:NewHurwitz}) in context. There is another variant, 
besides~\eqref{eq:Aphy} and~\eqref{eq:Aho}, of counting saddle connections. 
Given an affine invariant manifold~$\cMM$ we say that two saddle connections on
$(X,\omega) \in \cMM$ are {\em $\cMM$-parallel} if they are parallel
and stay parallel in a neighborhood of $(X,\omega)$ in~$\cMM$. 
(The terminology is completely analogous to the notion of {\em $\cMM$-parallel
cylinders} introduced in \cite{WrightCyl}.) We thus define 
the counting function $A^\Mp_{1 \lra 2}(T)$ and the Siegel-Veech constant
$c_{1 \lra 2}^\Mp (\cMM)$
in analogy to~\eqref{eq:Aho} and~\eqref{eq:defscSV}, counting once
every $\cMM$-parallel class of cylinders. \cite[Proposition~3.1]{emz}
can now be restated as $c_{1 \lra 2}^\Mp (\cMM) = c_{1 \lra 2}^\ho (\cMM)$
if $\cMM$ is a connected component of a stratum.
For Hurwitz spaces the two values can be different, but we will see
(Proposition~\ref{prop:admissible}) that their
difference becomes negligible as the degree of the covers tends to infinity.
\par
\medskip
In the first part of \cite{emz} on recursive computations of
Siegel-Veech constants, Eskin, Masur and Zorich called the locus of degenerate
surfaces that contribute to the Siegel-Veech counting the
{\em principal boundary}. At that time the notion of principal boundary 
was used only as a partial topological compactification. Presently, we 
dispose of a complete and geometric  compactification for the strata 
(\cite{strata}) and for Hurwitz spaces (by admissible covers), 
and we can then identify the principal boundary as part of the compactification 
(see \cite{chenPB} for the case of the strata and Section~\ref{sec:NewHurwitz} 
for the case of Hurwitz spaces).
The reader should keep in mind that the locus ``principal boundary''
depends on the type of saddle connections under consideration.
\par
\medskip
Finally we remark that there is a zoo of possibilities
of associating weights with saddle connections and cylinders and to define 
Siegel-Veech constants accordingly. This started with \cite{vorobets}, 
and see also \cite{baugou} for computations and conversions of Siegel-Veech
constants. 

