
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Volume recursion via intersection theory}
\label{sec:RelMZInt}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section we show that the two main theorems of the
introduction, Theorem~\ref{intro:IntFormula} and Theorem~\ref{intro:VolRec} 
are equivalent. This section does not yet provide a direct proof of
either of them. 
\par
We first show that the intersection numbers in Theorem~\ref{intro:IntFormula}
are given by a recursion formula of the same shape as in
Theorem~\ref{intro:VolRec}. Together with an agreement on the minimal
strata this proves the equivalence of the two theorems. Along the way
we introduce special classes of stable graphs that are used for recursions
throughout the paper. 

%%%%%%%%%%%%%%%%%%%%%
\subsection{Intersection numbers on the projectivized Hodge bundle}
\label{ssec:classes}
%%%%%%%%%%%%%%%%%%%%%

Fix $g$ and $n$ such that $2g-2+n>0$. We denote by $f\colon \cXX \to \barmoduli[g,n]$
the universal curve and  by $\omega_{\cXX/\barmoduli[g,n]}$ the relative dualizing line bundle.
We will use the following cohomology classes: 
\begin{itemize}
\item Let $1\leq i \leq n$. We denote by $\sigma_i\colon \barmoduli[g,n] \to \cXX$ the section of $f$ corresponding to the $i$-th marked point and by $\cLL_i=\sigma_i^* \omega_{\cXX/\barmoduli[g,n]}$ the cotangent line at the $i$-th marked point. With this notation, we define $\psi_i =c_1(\cLL_i)\in H^2(\barmoduli[{g,n}],\QQ)$.
\item For $1\leq i\leq g$, we denote by
$\lambda_i=c_i(\obarmoduli[{g,n}]) \in H^{2}(\barmoduli[{g,n}],\QQ)$
the $i$-th Chern class of the Hodge bundle. (We use the same notation for a vector bundle and its total space.) 
\item We denote by $\delta_0\in H^{2}(\barmoduli[{g,n}],\QQ)$ the Poincar\'e-dual class of the divisor parameterizing marked curves with at least one non-separating node.
\item The projectivized Hodge bundle $\proj\obarmoduli[{g,n}]$ comes with
the universal line bundle class $\xi=c_1(\cOO(1))\in H^2(\proj\obarmoduli[{g,n}],\QQ)$.
\end{itemize}
Unless otherwise specified, we denote by the same symbol a class
in $H^*(\oM_{g,n},\QQ)$ and its pull-back via the projection
$p\colon \proj\obarmoduli[{g,n}] \to \barmoduli[g,n]$. Recall that the splitting
principle implies that the structure of the cohomology ring of the
projectivized Hodge bundle is given by
$$
H^*(\proj\oOmM_{g,n},\Q) \,\simeq \,
H^*(\oM_{g,n},\Q)[\xi]/(\xi^g+\lambda_1 \xi^{g-1}+\cdots+\lambda_g)\,.
$$
\par
\medskip
Let $\mu=(m_1,\ldots,m_n)$ be a partition of $2g-2$. We denote by $\proj\obarmoduli[g,n](\mu)$ the closure of the projectivized
stratum $\proj\omoduli[g,n](\mu)$ inside the total space of
the projectivized Hodge bundle $\proj\obarmoduli[g,n]$.  This space is called the {\em (ordered) incidence variety compactification}\footnote{In \cite{strata}
the notation $\proj\obarmoduli[g,n]^{\textrm{inc}}(\mu)$ is used. 
Here we drop the superscript~``${\textrm{inc}}$'' for simplicity.
In \cite{SauvagetClass} this space is denoted by $\proj\oH_{g,n}(\mu)$.}.
\par
In this section we study the intersection numbers 
\begin{eqnarray}
\label{eq:a(mu)}
a_i(\mu) &=& \int_{\proj\obarmoduli[g,n](\mu)} \!\!\!\!\!\!\! \beta_i\cdot \xi =\frac{1}{m_i+1} \int_{\proj\obarmoduli[g,n](\mu)} \!\!\!\!\!\!\!  \xi^{2g-1} \cdot \prod_{j\neq i} \psi_j\, 
\end{eqnarray}
for all $1\leq i\leq n$. The reader should think of the~$a_i(\mu)$
as certain normalization of volumes. In fact, Theorem~\ref{intro:IntFormula} can be reformulated as
\be \label{eq:volai}
{\rm vol}(\omoduli[g,n](\mu)) \= \frac{2(2\pi)^{2g}(-1)^g}{(2g-3+n)!} \, 
a_i(\mu)\,,
\ee
implying in particular that $a_i(\mu)$ is independent of~$i$.
\par
We prove a collection of properties defining recursively the $a_i(\mu)$
 as the coefficients of some formal series. 
As the base case for $n=1$, i.e.~$\mu = (2g-2)$, define 
 the formal series 
\be \label{eq:defA}
\cAA(t) \= \frac{1}{t} +\sum_{g\geq 1} (2g-1)^2\, a_1(2g-2)  t^{2g-1} \quad
\in \frac{1}{t}\QQ[[t]]
\ee
and set 
\be \label{eq:defBandb}
B(z) \,:=\, \frac{z/2}{\sinh(z/2)} 
\, =: \, \sum_{j \geq 0} b_j z^j 
  \= 1 - \frac{z^2}{24} + \frac{7z^4}{5760} + \cdots\,.
\ee
For a partition $\mu$, recall that $n(\mu)$ denotes the number of its entries and 
$|\mu|$ denotes the sum of the entries. 
\par
\begin{Thm}\label{thm:indintall}
The  generating function $\cAA$ of the intersection numbers 
$a_i(2g-2)$ is determined by the coefficient extraction identity
\be \label{eq:ind1}
[t^{0}] \frac{1}{j!} \cAA(t)^{j} \= b_j \,,
%[t^{2g}] B(t) \= \frac{1}{ (2g)! } [t^{0}]  \cAA(t)^{2g} 
\ee
while the intersection numbers $a(\mu) = a_i(\mu)$ with $n(\mu) \geq 2$ are 
given recursively by
\begin{flalign} 
& \phantom{\=}
(m_1+1)(m_2+1) a(m_1,\ldots,m_n)  \label{eq:ind2} \\
&\=  \sum_{k=1}^{\min(m_1+1,m_2+1)} \!\! \frac{1}{k!}\, 
\sum_{\bfg, \bfmu}
h_{\proj^1}((m_1,m_2),\bfp) 
\prod_{j=1}^k (2g_j-1+n(\mu_j))\, p_j \, a(p_j-1,\mu_j)\,,
\nonumber
\end{flalign}
with the same summation conventions as in Theorem~\ref{intro:VolRec}.
\end{Thm}
\par
The first identity~\eqref{eq:ind1} was proved in~\cite{SauvagetMinimal}
and gives
\bes
\cAA(t) \= 
\frac{1}{t} - \frac1{24}t + \frac3{640}t^3 - \frac{1525}{580608}t^5
+ \frac{615881}{199065600}t^8 -\cdots\,.
\ees 
By Lagrange inversion, this formula can be written equivalently as
\bes
\cAA(t) \= \frac{1}{Q^{-1}(t)} \,, \quad \text{where} \quad
Q(u) \= u \, \exp\biggl(\sum_{k \geq 1} (k-1)! b_k u^k\biggr)
\ees 
%+ 615881/199065600*x^8 - 3058641/504627200*x^10 + 38800188510523/2191186722816000*x^12 - 3213747182969063/44497945755648000*x^14 + 100462329712125/255806104666112*x^16 - 43865443313064357090353257/15953645581139831685120000*x^18 + 4543042335221166932765440567147/188420950968830433165312000000*x^20 + O(x^21)
and will in fact be proved in this form in Section~\ref{sec:appvol}. We observe
in passing that $Q(u)$ is the asymptotic expansion of $\psi(u^{-1} + \tfrac12)$
as $u \to 0$, where $\psi(x) = \Gamma'(x)/\Gamma(x)$ is the digamma function.
The proof of the second identity~\eqref{eq:ind2} will be completed by the end 
of Section~\ref{ssec:treestrata}. 
\par
In the course of proving Theorem~\ref{thm:indintall} we will prove the
following complementary result, justifying the
implicitly used fact that $a_i(\mu)$ is independent of~$i$.
\par
\begin{Prop}\label{independent}
For all $1\leq i\leq n$, we have 
\begin{eqnarray*}
a_i(\mu) \= -  \int_{\proj\obarmoduli[g,n](\mu)} \!\!\!\!\!\!\!
  \xi^{2g-2} \cdot \prod_{j=1}^n \psi_j \,.
\end{eqnarray*}
\end{Prop}
\par
%%%%%%%%%%%%%%%%%%%%%
\subsection{Boundary components of moduli spaces of Abelian differentials}\label{ssec:boundary}
%%%%%%%%%%%%%%%%%%%%%

In Section~\ref{sec:hproj} we introduced several families of stable graphs
to describe the boundary of Hurwitz spaces. Here we show how these graphs
encode relevant parts of the boundary of moduli spaces of Abelian differentials. 
\par
The recursions in Theorem~\ref{intro:VolRec} and
Theorem~\ref{thm:indintall} can be phrased as sums over a small
subset of twisted level graphs, with only two levels and more constraints,
that we call (rational) backbone graphs, inspired by Figure~\ref{cap:BB}.
\par
\begin{figure}[ht]
\begin{tikzpicture}
\tikzstyle{every node}=[font=\scriptsize]

% Zeichnung No 1 % % % % % % % % % % % % % % % % % % % % % % % % % % % %
\fill (0,0) circle (1.5pt) node[right] {$X_5$};
\draw[](0,0) -- (45:2.45) node[above] {$X_4$}
	    (0,0) -- (75:1.8) node[above] {$X_3$}
	    (0,0) -- (105:1.8) node[above] {$X_2$}
	    (0,0) -- (135:2.45) node[above] {$X_1$};
\fill[]{(0,0) -- ++ (45:2.45) circle  (1.5pt)
	(0,0) -- (75:1.8) circle  (1.5pt)
	(0,0) -- (105:1.8) circle  (1.5pt)
	(0,0) -- (135:2.45) circle  (1.5pt)};
\draw[] (-2.5,.2) node[] {$\ell =-1$};
\draw[] (-2.5,1.7) node[] {$\ell =0$};

% Zeichnung No 2 % % % % % % % % % % % % % % % % % % % % % % % % % % % %
\draw[](3,-.2) -- (3,1.8) node[above] {$X_5$};
\draw[] (2.9,.5) -- (4.5,.5) node[right] {$X_3$}
        (2.9,1) -- (4.5,1)  node[right] {$X_2$};     
\path[draw]{(2.9,1.5) arc (117:90:1.75cm)  -- +(0:.8) node[right] {$X_1$}};
\path[draw]{(2.9,0) arc (-117:-90:1.75cm)  -- +(0:.8) node[right] {$X_4$}};
%\draw[] (4.5,-.3) -- (4.5,2); % Hilfslinie

\end{tikzpicture} 
\caption{A backbone graph and the corresponding stable curve} \label{cap:BB}
\end{figure}
\par
Recall that a bi-colored graph is a level graph with two levels $\{0, -1\}$ 
that has no horizontal edges. 
\par
\begin{Defi}
An  {\em almost backbone graph} is a bi-colored graph with only one vertex at
level~$-1$. For such a graph to be a {\em (rational) backbone graph} we
require moreover that it is of compact type and that the vertex at
level~$-1$ has genus zero.
\end{Defi}
\par
We denote by ${\rm BB}(g,n)\subset {\rm ABB}(g,n) \subset {\rm Bic}(g,n)$
the sets of backbone, almost backbone and bi-colored graphs.  We denote
by ${\rm BB}(g,n)_{1,2}\subset {\rm BB}(g,n)$ the set of backbone graphs such
that the first and second legs are adjacent to the vertex of level $-1$.
Moreover, let $\BB(g,n)^\star_{1,2}\subset {\rm BB}(g,n)_{1,2}$ be the subset
where precisely the first two legs are adjacent to the lower level vertex.
Similarly, we define $\ABB(g,n)_{1,2}$ and $\ABB(g,n)^\star_{1,2}$ and
drop $(g,n)$ if there is no source of confusion.
\par
We remark that the backbone graphs will play an important role here, while
the graphs in ${\rm ABB}(g,n)$ appear only in the Hurwitz space interlude in
Section~\ref{sec:NewHurwitz}. We fix some notations for these graphs,
used throughout in the sequel. For $\Gamma \in {\rm BB}(g,n)$ we
denote by $v_{-1}$ the vertex of level~$-1$. Given a
partition~$\mu = (m_1, \ldots, m_n)$ of $2g-2$, let $\bfp$ be the unique
twist of type $(\mu, \emptyset)$ for $\Gamma$ (see Definition~\ref{def:twist}).
We denote by $\mu[0]_{-1}$ the list of~$m_i$ for all legs~$i$ at level~$-1$
and with a slight abuse of notation we denote by $\bfp = (p_1,\ldots,p_k)$
the list of $\bfp(h)$ for half-edges $h$ that are adjacent
to the $k$ vertices of level~$0$.
Said differently, the restriction of the twist to level~$-1$ provides
$v_{-1}$ with a twist of type~$(\mu[0]_{-1}, \mu[\infty]_{-1} = \bfp)$. 
Finally if~$v$ is a vertex of level~$0$, we denote by $\mu_v$ the list
of $\bfp(h)-1$ for all half-edges adjacent to~$v$.
\par
\medskip
The goal in the remainder of the section is to introduce the classes
$\alpha_{\Gamma,\ell,p}$ in~\eqref{eq:defalphaGlI} below that will be used in
Proposition~\ref{prop:indclasses} to compute intersection numbers
on $\proj\obarmoduli[g,n](\mu)$. A stable graph  $\Gamma \in {\rm Stab}(g,n)$
determines the moduli space 
\begin{equation*}
\oM_\Gamma \= \prod_{v \in V(\Gamma)} \oM_{g(v),n(v)}\,
\end{equation*}
and comes with a natural morphism $\zeta_\Gamma\colon \oM_\Gamma \to  \oM_{g,n}$.
Let $\ell$ be a level function on $\Gamma$ such that $(\Gamma,\ell)$ is
a bi-colored graph with two levels $\{0, -1 \}$. We define the following 
vector bundle
$$
\oOmM_{\Gamma,\ell} \= \left(\prod_{v\in V(\Gamma), \ell(v)=0} \oOmM_{g(v),n(v)} \right)
\times  \left(\prod_{v\in V(\Gamma), \ell(v)=-1} \oM_{g(v),n(v)}\right)
$$
over $\oM_{\Gamma}$. This space comes with a natural morphism 
$\zeta_{\Gamma,\ell}^{\#}\,\colon\,\oOmM_{\Gamma,\ell} \to \oOmM_{g,n}$, defined by the 
composition $\oOmM_{\Gamma,\ell}\to \zeta_{\Gamma}^* \left(\oOmM_{g,n}\right) 
\to \oOmM_{g,n}$ where the first arrow is the inclusion of a vector sub-bundle 
and the second is the map on the  Hodge bundles induced from $\zeta_{\Gamma}$
by pull-back. The morphism  $\zeta_{\Gamma,\ell}^{\#}$ determines a 
morphism (denoted by the same symbol) on the projectivized Hodge bundles 
$\zeta_{\Gamma,\ell}^{\#}\,\colon\, \proj\oOmM_{\Gamma,\ell} \to \proj\oOmM_{g,n}\,$. The 
image of $\zeta_{\Gamma,\ell}^{\#}$ is the closure of the locus of differentials supported on curves with dual graph $\Gamma$ such that the differentials vanish identically on components of level~$-1$. In the sequel we will need the following lemma (see~\cite[Proposition~5.9]{SauvagetClass}).
\begin{Lemma}\label{lem:divisible}
 The Poincar\'e-dual class of $\zeta^{\#}_{\Gamma,\ell}(\proj\oOmM_{\Gamma,\ell})$ is divisible by $\xi^{h^1(\Gamma)}$ in $H^*(\proj\obarmoduli[g,n],\QQ)$.
\end{Lemma}
\par
Take a bi-colored graph $(\Gamma, \ell)$ and a partition $\mu$ 
of $2g-2$.
Now we consider a twist~$\bfp$ of type $(\mu[0] = \mu, \mu[\infty] = \emptyset)$
compatible with $\ell$ and
construct a subspace $\proj\oOmM_{\Gamma,\ell}^{\bfp}\subset  \proj\oOmM_{\Gamma,\ell}$
such that $\zeta_{\Gamma,\ell}^{\#}(\proj\oOmM_{\Gamma,\ell}^{\bfp})$ lies in the
boundary of $\proj\obarmoduli[g,n](\mu)$. Let 
\begin{eqnarray} \label{eq:m0m-1}
\OmM_{0} &\subset & \prod_{v\in V(\Gamma), \ell(v)=0} \OmM_{g(v),n(v)}\,, \nonumber\\
\M_{-1}&\subset& \prod_{v\in V(\Gamma), \ell(v)=-1} \mathcal M_{g(v),n(v)} 
\end{eqnarray}
be the loci defined by the following three conditions:
\begin{itemize}
\item[i)] A differential in $\OmM_{0}$ has zeros of orders $m_i$ at the relevant
  marked points and of orders $\bfp(h)-1$ at the relevant branches of the nodes.
\item[ii)] For each $v$ of level $-1$ there exists a non-zero (meromorphic) 
differential~$\omega_v$ on the component $X_v$ corresponding to $v$ that has
zeros at the relevant marked points of orders prescribed by $\mu$ and poles 
at the relevant branches of the nodes of orders prescribed by~$\bfp$, i.e.\
such that the canonical divisor class of $X_v$ is given by
$\sum_{h\in H, a(h)=v} (\bfp(h)-1)x_h$, where $x_h \in X_v$ is the marked point 
or the node corresponding to the half-edge~$h$. 
\item[iii)] There exist complex numbers $k_v\neq 0$ for all vertices $v$ of level $-1$ such that $\omega=\sum_{\ell(v) = -1} k_v \omega_v$ satisfies the global residue condition of~\cite{strata}.  
\end{itemize}
In particular for a backbone graph $\Gamma$, since it is of compact type with a unique vertex $v_{-1}$ of level $-1$, we have the identification 
$\OmM_0=\prod_{v\neq v_{-1}} \OmM_{g(v),n(v)}(\mu_v)$. 
\par
We define $\proj\oOmM_{\Gamma,\ell}^{\bfp}$ as the Zariski closure of
$\proj \OmM_{0} \times \M_{-1}$ in $\proj\oOmM_{\Gamma,\ell}$ and define  
\be \label{eq:defalphaGlI}
 \alpha_{\Gamma,\ell,\bfp} = \left\{ \begin{array}{cl}
   \zeta_{\Gamma,\ell*}^{\#} [\proj\oOmM_{\Gamma,\ell}^{\bfp}] & \text{if }
   \dim(\proj\oOmM_{\Gamma,\ell}^{\bfp})=\dim(\proj\obarmoduli[g,n](\mu))-1 \\
0& \text{otherwise}
\end{array} 
\right.
\ee
as the corresponding class in $H^*(\proj\obarmoduli[g,n],\QQ)$.
By~\cite[Proposition~5.9]{SauvagetClass}, we can describe $\alpha_{\Gamma,\ell,\bfp}$
with the following lemma. 
\par
\begin{Lemma}\label{lem:divisible2}
If  $(\Gamma,\ell,\bfp)$ is a bi-colored graph of compact type, then
$\alpha_{\Gamma,\ell,\bfp}\neq 0$ if and only if there is a unique vertex  $v_{-1}$ of
level $-1$, and in this case $\alpha_{\Gamma,\ell,\bfp}$ is divisible by
$$
\xi^{g(v_{-1})} + \zeta_{\Gamma*}(\lambda_{v_{-1},1}) \xi^{g(v_{-1})-1} +\cdots +  \zeta_{\Gamma*}(\lambda_{v_{-1},g(v_{-1})})\,,
$$
where 
\begin{eqnarray*}
\lambda_{v_{-1},i}=(\lambda_i, 1,\ldots, 1) & \in & H^*(\oM_{g(v_{-1}),n(v_{-1})},\QQ) \!\!\!\bigotimes_{v\in V(\Gamma), v\neq v_{-1}} \!\!\! H^*(\oM_{g(v),n(v)},\QQ) \\
&\simeq & H^*(\oM_{\Gamma},\QQ)\,.
\end{eqnarray*}
\end{Lemma}

%%%%%%%%%%%%%%%%%%%%%
\subsection{A first reduction of the computation }\label{ssec:reduction}
%%%%%%%%%%%%%%%%%%%%%

Recall the (marked and projectivized) Hodge bundle projection  
$p\colon \proj\obarmoduli[{g,n}] \to \barmoduli[g,n]$.  As before we usually denote by the same symbol a class in  $\barmoduli[g,n]$ and its pullback via $p$.  
In this section we show that many $p$-push forwards of intersections
of $\alpha_{\Gamma,\ell,p}$ with tautological classes vanish or can
be computed recursively. The starting point is the following important
lemma proved by Mumford in~\cite[Equation (5.4)]{mumford83}.
\par
\begin{Lemma} \label{lemMum}
The Segre class of the Hodge bundle is the Chern class
of the dual of the Hodge bundle, i.e.\ 
$$
c_*(\oOmM_{g,n})\cdot c_*(\oOmM_{g,n}^{\vee}) \= 1\,.
$$
In particular, we have $\lambda_g^2=0\in H^{4g}(\oM_{g,n},\QQ)$.
\end{Lemma}
\par
Together with the definition of Segre class, this lemma implies that 
$$
p_*(\xi^{k} \gamma) \= s_{k-g+1}(\oOmM_{g,n}) \gamma \= 
(-1)^{k-g+1} \lambda_{k-g+1} \gamma  
$$
for all $\gamma\in H^*(\oM_{g,n},\QQ)$ and all $k\geq g-1$. Another 
important lemma is the following (see e.g.~\cite[Section 13, 
Equation~(4.31)]{acgh2}).
\par
\begin{Lemma}\label{lem:intlambda} 
Let $1\leq k\leq g$ and let $\Gamma$ be a stable graph. Then 
$$\zeta_{\Gamma}^* \lambda_{k} \=
\sum_{(k_v)_{v\in V} \in \NN^V , \atop |(k_v)|=k} \, \prod_{v\in V} \lambda_{k_v}\,,$$
where the sum is over all partitions of $k$ into non-negative integers $k_v$ 
assigned to each vertex $v \in V = V(\Gamma)$. 
\end{Lemma}
\par
In particular if $h^1(\Gamma)> g-k$, then $\sum_{v\in V}g(v) =
g - h^1(\Gamma) < k$, hence the above lemma implies that 
$\zeta_{\Gamma}^* \lambda_{k} = 0$ as there exists some $k_v > g(v)$
for any partition $(k_v)_{v\in V}$ of $k$.  
\par
As a consequence of the above discussion, we obtain the following result.
\begin{Lemma}\label{lem:toplambda}
Let $\alpha=\sum_{i\geq 0}\xi^i \alpha_i$ be a class in $H^*(\proj \oOmM_{g,n},\QQ)$
where the classes $\alpha_i$ are pull-backs from $H^*(\oM_{g,n},\Q)$. Then we have
\begin{eqnarray*}
p_*(\xi^{2g-1} \alpha)&=&(-1)^{g} \alpha_0 \lambda_g\,, \\
p_*(\xi^{2g-2} \alpha)&=&  (-1)^{g} \alpha_1 \lambda_g +(-1)^{g-1} \alpha_0  \lambda_{g-1}\, ,\\
p_*(\xi^{2g-2} \delta_0\alpha)&=&  (-1)^{g-1} \alpha_0 \delta_0 \lambda_{g-1}\,.
\end{eqnarray*}
\end{Lemma}
\par
Recall the expressions of the intersection numbers $a_i(\mu)$ in~\eqref{eq:a(mu)} and in Proposition~\ref{independent}. 
In order to compute $a_i(\mu)$, by Lemma~\ref{lem:toplambda}
we only need to consider the $\xi$-degree zero and one parts of the class $[\proj\oOmM_{g,n}(\mu)]$ in $H^*(\proj \oOmM_{g,n},\QQ)$.  
\par
Combining Lemmas~\ref{lem:intlambda} and~\ref{lem:toplambda} together
with Lemmas~\ref{lem:divisible} and~\ref{lem:divisible2} of the previous
section, we can already prove the following vanishing result for classes
associated with some bi-colored graphs.
\par
\begin{Prop}\label{pr:intbb}
If $(\Gamma,\ell,\bfp)$ is not a backbone graph, then 
$$p_{*}\left(\xi^{2g-2}\alpha_{\Gamma,\ell,\bfp}\right) \= 0 \in H^*(\oM_{g,n},\QQ)\,
$$
where $\alpha_{\Gamma,\ell,\bfp}$ is defined in~\eqref{eq:defalphaGlI}. 
\end{Prop}
\par
\begin{proof}
For simplicity we write $\alpha = \alpha_{\Gamma,\ell,\bfp}$ in the proof.  
We assume first that $\Gamma$ is not of compact type, i.e. $h^1(\Gamma) > 0$.
Then by Lemma~\ref{lem:divisible}, the class $\alpha $ is divisible
by~$\xi$. Note that the cohomology ring of a projective bundle is
generated by the universal line bundle class with the classes pulled
back from the base. Therefore, we can write 
$$\xi^{2g-2}\alpha \=\sum_{i\geq 0} \xi^{2g-1+i} \alpha_i'\,,$$
where $\alpha_i'$ is a pullback from $H^*(\oM_{g,n},\QQ)$ that is
supported on $\zeta_\Gamma(\oM_{\Gamma})$ for all $i\geq 0$. Thus
by Lemma~\ref{lem:toplambda}, $p_*(\xi^{2g-2}\alpha)= (-1)^{g} \alpha_0'
\lambda_g=0$, because $\Gamma$ is not of compact type.
\par
Now we assume that $\Gamma$ is of compact type. By Lemma~\ref{lem:divisible2}, we only need to consider the case when there is a unique vertex $v_1$ of level $-1$. 
Since $\Gamma$ is not a backbone graph, $v_1$ has positive genus $g_1$. Still by Lemma~\ref{lem:divisible2} and simplifying the notation ${\zeta_{\Gamma}}_{*}(\lambda_{v_1,i})$ 
by $\lambda_{v_1,i}$, the class $\alpha$ is divisible by $\xi^{g_1}+ \xi^{g_1-1}\lambda_{v_1,1} + \cdots+ \lambda_{v_1,g_1}$. Consequently we can write 
$$
\alpha \= ( \xi \lambda_{v_1,g_1-1}+\lambda_{v_1,g_1}) \gamma_0 \,+\,
\xi \lambda_{v_1,g_1} \gamma_1 \,+ \,O(\xi^2)\,,
$$
where $\gamma_0$ and $\gamma_1$ are pullbacks from $H^*(\oM_{g,n},\QQ)$
and the $O(\xi^2)$ term stands for a class divisible by $\xi^2$. By
Lemma~\ref{lem:toplambda}, we obtain that 
$$
p_*(\xi^{2g-2}\alpha) \= (-1)^{g} (\lambda_{v_1,g_1-1} \lambda_{g}- \lambda_{v_1,g_1} \lambda_{g-1}) \gamma_0 + (-1)^{g} \lambda_{v_1,g_1} \lambda_{g} \gamma_1\,. 
$$
Using Lemma~\ref{lem:intlambda}, we also obtain that  
\begin{eqnarray*}
\zeta_{\Gamma}^*(\lambda_g) \= \bigotimes_{v \in V} \lambda_{g_v} \quad \text{ and } \quad 
\zeta_{\Gamma}^*(\lambda_{g-1}) \=\sum_{v \in V} \left( \lambda_{g_v-1}  \bigotimes_{v' \neq v} \lambda_{g_{v'}} \right).
\end{eqnarray*}
From the projection formula we deduce that 
$$\lambda_{v_1,g_1}\cdot \lambda_g \= {\zeta_{\Gamma}}_{*} (\lambda_{v_1,g_1}\cdot \zeta^{*}_{\Gamma}(\lambda_g)) \= 
\zeta_{\Gamma_*}\left(\lambda_{g_1}^2 \bigotimes_{v' \neq v_1} \lambda_{g_{v'}} \right) \=0\,,$$
because $\lambda_{g_1}^{2}=0 \in H^*(\oM_{g(v_1),n(v_1)},\QQ)$ by Lemma~\ref{lemMum}.
Once again the same lemma implies that 
\begin{eqnarray*}
\lambda_{v_1,g_1-1}\cdot \lambda_g&=&  \zeta_{\Gamma_*}\left(\lambda_{g_1}\lambda_{g_1-1} \bigotimes_{v' \neq v_1} \lambda_{g_{v'}} \right), \\
\lambda_{v_1,g_1}\cdot \lambda_{g-1}&=&\zeta_{\Gamma_*}\left(\lambda_{g_1}\lambda_{g_1-1} \bigotimes_{v' \neq v_1} \lambda_{g_{v'}} \right) + \sum_{v \neq v_1}\sum_{v'\neq v, v_1}  \left( \lambda_{g_1}^2 \otimes \lambda_{g_v-1} \bigotimes_{v' \neq v} \lambda_{g_{v'}} \right)\\
&=&\lambda_{v_1,g_1-1}\cdot \lambda_g\,.
\end{eqnarray*}
Putting everything together, we thus conclude that $p_*(\xi^{2g-2}\alpha)=0$.
\end{proof}
\par
We define the {\em multiplicity} of a twist $\bfp$ to be
\be \label{eq:multtwist}
m(\bfp) \= \prod_{(h,h')\in E(\Gamma)} \sqrt{-\bfp(h)\bfp(h')}\,.
\ee
\par
\begin{Prop}\label{pr:intbb1}
If $(\Gamma,\ell,\bfp)$ is a backbone graph in  $\BB(g,n)_{1,2}$, then 
\bes
\int_{\proj\oOmM_{g,n}} \!\!\!\!\!\alpha_{\Gamma,\ell,\bfp} \cdot \xi^{2g-1} \cdot
\prod_{i=3}^n \psi_i  \= {m(\bfp)} \cdot  h_{\proj^1}(\mu_{-1},\bfp)
\cdot \prod_{v\in V(\Gamma) \atop \ell(v)=0} a_1({p_v-1,\mu_v})\,,
\ees
where $p_v$ is the entry of $\bfp$ corresponding to the twist on the unique edge
of each vertex~$v$ of level $0$ and $\mu_{-1}$ is the list of entries in $\mu$ whose corresponding legs are adjacent to the vertex of level $-1$. 
\end{Prop}
\par
As a preparation for the proof we relate the space $\M_{-1}$ defined 
in~\eqref{eq:m0m-1} to the Hurwitz space for backbone graphs. The idea 
behind this relation was already mentioned in the last paragraph of 
Section~\ref{subsec:n=2}. If $(\Gamma,\ell)$ is a backbone graph, then we 
claim that $H_{\proj^1}(\mu_{-1},\bfp)\cong \M_{-1}$,
where the isomorphism is provided by the source map $f_S$ that marks the critical points of the branched covers. To verify the claim, let $\omega$ be the meromorphic differential on the unique vertex of $\Gamma$ of level $-1$ as in part iii) of the definition for $\M_{-1}$. Since $\Gamma$ is of compact type, 
the global residue condition in~\cite{strata} imposed to $\omega$ implies that all residues of $\omega$ vanish. Therefore, a point in $\M_{-1}$ can be identified with such a meromorphic differential $\omega$ (up to scale)  
on $\proj^1$ without residues, such that $\omega$ has zeros of order $m_i$ for $m_i \in \mu_{-1}$ at the corresponding markings and has poles of order $p_j+1$ 
for $p_j \in \bfp$ at the corresponding nodes. In particular, $\omega$ is an exact differential and integrating it on $\proj^1$ 
provides a meromorphic function that can be regarded as a branched cover $f$ parameterized in $H_{\proj^1}(\mu_{-1},\bfp)$.  Conversely given 
$f$ in $H_{\proj^1}(\mu_{-1},\bfp)$, we can treat $f$ as a meromorphic function and taking $df$ gives rise to such $\omega$.  We thus conclude that 
 $H_{\proj^1}(\mu_{-1},\bfp)\cong \M_{-1}$. 
Consequently for $\Gamma \in \BB(g,n)_{1,2}$, we have 
\begin{equation}\label{eqn:m1}
h_{\proj^1}(\mu_{-1},\bfp) \= \int_{\BH_{\proj^1}(\mu_{-1},\bfp)} f_S^*
\Biggl(\prod_{\begin{smallmatrix} 3\leq i\leq n\\  i\mapsto v_{-1} \end{smallmatrix}}
\psi_i\Biggr) \= \int_{\oM_{-1}} \prod_{\begin{smallmatrix} 3\leq i\leq n\\
i\mapsto v_{-1} \end{smallmatrix}}  \psi_i\,,
\end{equation} 
where $i\mapsto v_{-1}$ means that the $i$-th marking belongs to the vertex of
level $-1$. 
\par
Now we can proceed with the proof of Proposition~\ref{pr:intbb1}. 
\begin{proof}
We write $\alpha_{\Gamma,\ell, \bfp}=\sum_{i\geq 0} \alpha_{\Gamma,\ell, \bfp}^i \xi^i$
where $\alpha_{\Gamma,\ell, \bfp}^i$ is a pull-back from $H^*(\oM_{g,n},\QQ)$.
By Lemma~\ref{lem:toplambda} we deduce that 
\ba \label{eq:pushfwd}
p_*(\xi^{2g-1}\alpha_{\Gamma,\ell, p}) &\= (-1)^{g} \lambda_g \alpha_{\Gamma,\ell, \bfp}^0\,.
%\\
%p_*(\xi^{2g-2}\delta_0\alpha_{\Gamma,\ell, \bfp}) &\=
%(-1)^{g-1} \lambda_{g-1} \alpha_{\Gamma,\ell, \bfp}^0 \delta_0\,.
\ea 
Therefore we only need to consider the $\xi$-degree zero part of
$\alpha_{\Gamma,\ell, \bfp}$, which is given by
\begin{eqnarray*}
\zeta_{\Gamma*}\Bigl( [\oM_{-1}] \bigotimes_{v \in V(\Gamma), \ell(v)=0} [\proj \oOmM_{g_v,n_v}(p_v-1,\mu_v)]^0  \Bigr),
\end{eqnarray*} 
where 
$[\proj \oOmM_{g_v,n_v}(p_v-1,\mu_v)]^0$ is the degree zero part of the
Poincar\'e-dual class of $\proj \oOmM_{g_v,n_v}(p_v-1,\mu_v)$ in
$\proj \oOmM_{g_v,n_v}$. Therefore, we have
\begin{eqnarray*}
\lambda_g \cdot \alpha_{\Gamma,\ell,\bfp}^0 
\= \zeta_{\Gamma*}\left( [\oM_{-1}] \bigotimes_{v \in V(\Gamma), \ell(v)=0} 
\left( \lambda_{g_v} \cdot [\proj \oOmM_{g_v,n_v}(p_v-1,\mu_v)]^0 \right)\right)\,.
\end{eqnarray*} 
Multiplying this expression by $\prod_{i=3}^n \psi_i$, we obtain that 
\begin{eqnarray*}
\lambda_g  \cdot \alpha_{\Gamma,\ell,\bfp}^0\cdot \prod_{i=3}^n \psi_i  \=&& \!\!\!\!\! \!\!\!\!\!  \zeta_{\Gamma*}\Bigg( \Big([\oM_{-1}] \cdot\prod_{i \mapsto v_{-1}, i\geq 3} \psi_i \Big) \\ && \bigotimes_{v \in V(\Gamma), \ell(v)=0} \left( \lambda_{g_v} \cdot [\proj \oOmM_{g_v,n_v}(p_v-1,\mu_v)]^0  \cdot \prod_{i \mapsto v} \psi_i \right) \Bigg)\,.
\end{eqnarray*}
For the first factor on the right-hand side, equality~\eqref{eqn:m1} implies that 
$$[\oM_{-1}] \cdot \prod_{i \mapsto v_{-1}, i\geq 3}\psi_i
\= h_{\proj^1}(\mu_{-1}, \bfp)\,.$$ 
Moreover for all $v$ of level $0$, we have
\begin{eqnarray*}
{p_v \,a_1(p_v-1,\mu_v)}&=& \int_{\proj \oOmM_{g_v,n_v}(p_v-1,\mu_v)}  \!\!\!\!\! \!\!\!\!\!  \xi^{2g_v-1} \prod_{i \mapsto v} \psi_i \\
&=& (-1)^{g_v} \int_{\oM_{g_v,n_v}} \!\!\!\!\!   \lambda_{g_v} \cdot \prod_{i \mapsto v} \psi_i\cdot [\proj \oOmM_{g_v,n_v}(p_v-1,\mu_v)]^0\,
\end{eqnarray*}
where the second identity follows from Lemma~\ref{lem:toplambda}. Since $p_v$ is the (positive) twist value assigned to the edge of $v$, the product of $p_v$ over all vertices of level $0$ equals    
$m(\bfp)$ defined in~\eqref{eq:multtwist}. In addition, the sum of $g_v$ over all vertices of level $0$ equals the total genus $g$, because $\Gamma$ 
is of compact type and $v_{-1}$ has genus zero.  Putting everything together we thus obtain that
$$
\int_{\oM_{g,n}}  \!\!\!\!\!  \lambda_g \cdot \alpha_{\Gamma,\ell,\bfp}^0 \cdot \prod_{i=3}^n \psi_i= m(\bfp)\cdot {h_{\proj^1}(\mu_{-1}, \bfp)}\cdot\!\!\!\!\!\!\!  \!\!\!\!\!  \prod_{v\in V(\Gamma),\ell(v)=0}\!\!\!\!\! a_1({p_v-1,\mu_v})
\,,
$$
which is the desired statement.
\end{proof}
\par

%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\subsection{The induction formula for cohomology classes}\label{ssec:induction}
\label{sec:indCC}
%%%%%%%%%%%%%%%%%%%%%%%%%%%5

The main tool of the section is the induction formula 
in~\cite[Theorem~6 (1)]{SauvagetClass} which we recall now.
\par
\begin{Prop}\label{prop:indclasses}
For all $1\leq i\leq n$, the relation that 
\be \label{for:indclasses}
\left(\xi+(m_i+1)\psi_i\right)[\proj\obarmoduli[g,n](\mu)]
\= \sum_{\begin{smallmatrix}(\Gamma,\ell,p)\\ i\mapsto v, \ell(v)=-1\end{smallmatrix}}
\frac{m(\bfp)}{|{\rm Aut}(\Gamma,\ell,\bfp)|}  \alpha_{\Gamma,\ell,\bfp}
\ee
holds in $H^*(\proj\obarmoduli[g,n],\QQ)$, where the sum is over all
twisted bi-colored graphs such that the $i$-th leg is carried by a
vertex of level $-1$.
\end{Prop}
There are two ways of using equation~\eqref{for:indclasses}.
First one can compute the Poincar\'e-dual class of $\proj\obarmoduli[g,n](\mu)$ in $H^*(\proj\obarmoduli[g,n],\QQ)$ in terms of the $\psi$, $\lambda$, $\xi$ classes and boundary classes associated to stable graphs. This strategy is used in~\cite{SauvagetMinimal} to deduce the first formula in Theorem~\ref{thm:indintall}.
\par
Alternatively, one can compute relations in the Picard group
of $\proj\obarmoduli[g,n](\mu)$ to deduce relations between intersection numbers on $\proj\obarmoduli[g,n](\mu)$ and intersection numbers on boundary strata associated to twisted graphs. This is the strategy that we will use here. We will use this proposition with $i \in \{1,2\}$ and multiply the formula by $\xi^{2g-1} \prod_{i=3}^n \psi_i$ 
to obtain $a_1(\mu)$  
on the left-hand side. Then we will use
Propositions~\ref{pr:intbb} and~\ref{pr:intbb1} to compute the right-hand
side. A first application of this strategy gives a proof of the
complementary proposition. 
\par
\begin{proof}[Proof of Proposition~\ref{independent}] 
We use Proposition~\ref{prop:indclasses} with $i=1$. Multiplying formula~\eqref{for:indclasses} by $\xi^{2g-2}\cdot \prod_{i=2}^n \psi_i$, we obtain that 
\begin{eqnarray*}
(m_1+1) \left(a_1(\mu)+ \int_{\proj\obarmoduli[g,n](\mu)} \!\!\!\!\!\!\!  \xi^{2g-2} \cdot \prod_{i=1}^n \psi_i\right) &=&\\
 \sum_{\begin{smallmatrix}(\Gamma,\ell,\bfp)\\ 1\mapsto v, \ell(v)=-1\end{smallmatrix}} \!\!\!\!\!\! \frac{m(\bfp)}{|{\rm Aut}(\Gamma,\ell,\bfp)|} \!\!\!\!\!\! && \!\!\!\!\!\! \int_{\proj\obarmoduli[g,n]} \!\!\!\!\!\! \alpha_{\Gamma,\ell,\bfp} \cdot \xi^{2g-2} \cdot \prod_{i=2}^n \psi_i\,.
\end{eqnarray*}
It suffices to check that each summand in the right-hand side vanishes.
Proposition~\ref{pr:intbb} implies that if $(\Gamma,\ell,\bfp)$ is not a
backbone graph, then the corresponding summand vanishes. If $(\Gamma,\ell,\bfp)$
is a backbone graph, then we have seen (in the paragraph below Proposition~\ref{pr:intbb1}) that $\oM_{-1}$ is birational to a Hurwitz space of admissible covers of dimension $n_{-1}-2$,
where $n_{-1}$ is the number of legs adjacent to the vertex of level~$-1$. Since the $\psi$-product restricted to level $-1$ contains $n_{-1}-1$ terms (i.e. it misses $\psi_1$ only), which is bigger than $\dim \oM_{-1}$, it implies that the intersection of $\alpha_{\Gamma,\ell,\bfp}$ with $\prod_{i=2}^n\psi_i$
vanishes.  
\end{proof}
\par
Now we know that $a_i(\mu)$ is independent of the choice
of $1\leq i\leq n$ and hence we can drop the subscript~$i$.
The second use of the strategy presented above leads to the following
induction formula.
\par
\begin{Lemma}\label{lem:indbis}
The intersection numbers $a(\mu)$ satisfy the recursion 
\begin{eqnarray*}
  (m_1+1)(m_2+1) a(\mu)  \=
\sum_{ k \geq 1}
\sum_{\bfg, \bfmu} 
h_{\PP^1}((m_1,m_2,\mu_0),\bfp) \cdot \frac{1}{k!}
\cdot \prod_{i=1}^k p_i^2 a(p_i-1,\mu_i)\,
\end{eqnarray*}
where $\bfg=(g_1,\ldots,g_k)$ is a partition of $g$, $\bfmu = (\mu_0, \mu_1,\ldots,\mu_k)$ is a $(k+1)$-tuple of multisets with 
$(m_3,\ldots, m_n)= \mu_0\sqcup \cdots \sqcup  \mu_k$ and $\bfp = (p_1, \ldots, p_k)$ has entries $p_i = 2g_i - 1 - |\mu_i| > 0$.
\end{Lemma}
\par
We remark that this induction formula is not quite the same as the induction formula
of Theorem~\ref{thm:indintall}, e.g.\ the sums in the two formulas do not run over the
same set. Theorem~\ref{thm:indintall} will follow further from a combination
of Lemma~\ref{lem:indbis} and Proposition~\ref{prop:intpsiPP1} of the previous section.
\begin{proof}
We apply the induction formula of Proposition~\ref{prop:indclasses} with $i=2$:
$$
\left(\xi+(m_2+1)\psi_2\right)[\proj\obarmoduli[g,n](\mu)]= \sum_{\begin{smallmatrix}(\Gamma,\ell,\bfp)\\ 2\mapsto v, \ell(v)=-1\end{smallmatrix}}\frac{m(\bfp)}{|{\rm Aut}(\Gamma,\ell,\bfp)|}  \alpha_{\Gamma,\ell,\bfp}\,.
$$
We multiply this expression by $\xi^{2g-1}\prod_{i=3}^n \psi_i$ and apply $p_*$.
Since Lemma~\ref{lemMum} gives $p_*(\xi^{2g}  [\proj\obarmoduli[g,n](\mu)])=0$,
the above equality implies that  
$$
(m_1+1)(m_2+1)a(\mu) \= \sum_{\begin{smallmatrix}(\Gamma,\ell,\bfp)\\ 2\mapsto v, \ell(v)=-1\end{smallmatrix}}\frac{m(\bfp)}{|{\rm Aut}(\Gamma,\ell,\bfp)|} p_*\left(\xi^{2g-1} \cdot \prod_{i=3}^n \psi_i \cdot \alpha_{\Gamma,\ell,\bfp}\right)\,.
$$
By Proposition~\ref{pr:intbb} a term in the sum of the
right-hand side vanishes if $(\Gamma,\ell,p)$ is not a backbone graph. Suppose
$(\Gamma,\ell,\bfp)$ is a backbone graph such that 
the first leg does not belong to the vertex of level $-1$ (which contains $n_{-1}$ legs). Then on level $-1$ the product of $\psi$-classes contains $n_{-1}-1$ terms (i.e.\ this product misses $\psi_2$ only), which exceeds the dimension of $\oM_{-1}$ (being $n_{-1}-2$), hence the corresponding term in the sum also vanishes. 
\par
Now we only need to consider the case when $(\Gamma, \ell,\bfp)$ is a backbone graph
in ${\rm BB}(g,n)_{1,2}$, i.e.\ the vertex of level $-1$ carries both the first and
second legs.   Then the intersection number
$\xi^{2g-1}\cdot\prod_{i=3}^n \psi_i \cdot \alpha_{\Gamma,\ell,\bfp}$ is given
by Proposition~\ref{pr:intbb1}. We thus conclude that  
\begin{eqnarray*}
 (m_1+1)(m_2+1) a(\mu)  &\=&  \!\!\!\!\!\!
\sum_{(\Gamma, \ell, \bfp)\in {\rm BB}_{1,2}}  \frac{h_{\proj^1}(\mu_{-1}, \bfp) \,m(\bfp)^2 }{|{\rm Aut}(\Gamma,\ell,\bfp)|}\\
& & \quad  \cdot  \prod_{v\in V(\Gamma),\ell(v)=0}\!\!\!\!\! a({p_v-1,\mu_v})\\
&\=&  % \!\!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! 
\!\!\!\!\!  \sum_{k \geq 1} \sum_{\bfg, \bfmu}
m(\bfp)^2 h_{\PP^1}((m_1,m_2,\mu_0),\bfp) \cdot \frac{1}{k!}
\cdot \prod_{i=1}^k  a(p_i-1,\mu_i)\,.
\end{eqnarray*}
The last equality comes from the fact that the datum of $g_1+\cdots +g_k=g, \,g_i \geq 1$ and  $(m_3,\ldots, m_n)= \mu_0 \sqcup \mu_1\sqcup \cdots \sqcup  \mu_k$ determines uniquely a graph $(\Gamma, \ell,\bfp)$ in ${\rm BB}(g,n)_{1,2}$ and an automorphism of the backbone graph is determined by a permutation in $S_k$ that preserves both the partition of $g$ and the sets $\mu_1,\ldots,\mu_k$. 
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Sums over rooted trees}\label{ssec:treestrata}
%%%%%%%%%%%%%%%%%%%%%%%%%%%

The purpose of this section is to combine the preceding Lemma~\ref{lem:indbis}
with Proposition~\ref{prop:intpsiPP1} that describes the computation of
intersection numbers on Hurwitz spaces. We will show that the numbers
$a(\mu)$ can be expressed as sums over rooted trees in a similar way
as we did for intersection numbers on Hurwitz spaces in Section~\ref{sec:HurRT}.
\par
Let $2\leq i\leq n$ and $(\Gamma,\ell,\bfp)$ be a rooted tree in ${\rm RT}(g,\mu)_{1,i}$
(here $\mu[\infty]$ is empty). Since there is no marked pole, it implies that 
any vertex of genus zero has at least one edge with a negative twist, hence it is an internal vertex of $\Gamma$ and lies on a negative level.  
Denote by $\mu[\infty]_0$ the list obtained by taking the (positive) entries $\bfp(h)$
for all half-edges~$h$ adjacent to a vertex of level~$0$.   
Denote by $\mu[0]_0$ the list of entries of $\mu$ from those legs carried by the internal vertices (of genus zero). 
With this notation we define the rooted tree $(\Gamma_0,\ell_0, \bfp_{0})$ in ${\rm RT}(0,\mu[0]_0,\mu[\infty]_0)_{1,i}$ obtained by removing the leaves of $\Gamma$ (i.e. vertices of positive genus and hence on level $0$). We also define the multiplicity $m_0(\bfp)$ of $(\Gamma_0,\ell_0, \bfp_{0})$ to be the product of entries of $\mu[\infty]_0$. Now we define the $a$-contribution of the rooted tree $(\Gamma,\ell,\bfp)$ as
\begin{eqnarray}\label{eq:a-contribution}
a(\Gamma,\ell,\bfp)&=& m_0(\bfp)^2 h(\Gamma_0,\ell_0,\bfp_{0})
\prod_{v\in V(\Gamma),\ell(v)=0}\!\!\!\!\! a({p_v-1,\mu_v})\,,
\end{eqnarray}
where $h(\Gamma_0,\ell_0,\bfp_{0})$ is the contribution of the rooted tree defined in~\eqref{eq:h(gamma)}.   
\par
\begin{Lemma}\label{lem:inttree} The following equality holds: 
$$
(m_1+1)(m_2+1) a(\mu) \= \sum_{(\Gamma,\ell, \bfp)\in {\rm RT}(g,\mu)_{1,2}} \frac{a(\Gamma,\ell,\bfp)}{|{\rm Aut}(\Gamma,\ell,\bfp)|}\,.
$$
\end{Lemma}
\par
\begin{proof}
Removing the leaves of a rooted tree induces a bijection
between ${\rm RT}(g,\mu)_{1,2}$ and the set 
$$\bigcup_{(\Gamma,\ell, \bfp) \in {\rm BB}_{1,2}} {\rm RT}(0,\mu[0]_0,\mu[\infty]_0)_{1,2}
$$
which is a partition of ${\rm RT}(g,\mu)_{1,2}$ over all possible decorations
of the leaves of the rooted trees (i.e. each decoration is induced by a graph
in ${\rm BB}_{1,2}$).  Moreover, an automorphism of a rooted tree in
${\rm RT}(g,\mu)_{1,2}$ is determined by an automorphism of the backbone graph
in ${\rm BB}(g,n)_{1,2}$, because all internal vertices of the rooted tree (i.e.
those of genus zero and hence on negative levels) have marked legs by
Definition~\ref{def:RT}.  Then we can first use Lemma~\ref{lem:indbis} to
write $a(\mu)$ as a sum over backbone graphs in ${\rm BB}(g,n)_{1,2}$ and
then use Proposition~\ref{prop:intpsiPP1} to express it as the desired sum over the set
$$\bigcup_{(\Gamma,\ell, \bfp) \in {\rm BB}_{1,2}}
{\rm RT}(0,\mu[0]_0,\mu[\infty]_0)_{1,2}\,\simeq\, {\rm RT}(g,\mu)_{1,2}$$
as claimed in the lemma.
\end{proof}
\par
We define 
$$
{\rm RT}(g,\mu)_1= \{ \text{trivial graph}\} \,\cup \,
\bigcup_{i=2}^n {\rm RT}(g,\mu)_{1,i}$$
and the $a$-contribution of the trivial graph~$\bullet$ as
$a(\bullet,\ell, \bfp) = (m_1+1)^2 \, a(\mu)$.
\par
\begin{proof}[End of the proof of Theorem~\ref{thm:indintall}]
We will prove for $n\geq 2$ the equality that 
\begin{align}
\sum_{k \geq 1} \sum_{\bfg, \bfmu}
h_{\proj^1}((m_1,m_2),\bfp)\, \frac{1}{k!}\,&
\prod_{j=1}^k (2g_j-1+n(\mu_j)) p_j a(p_j-1,\mu_j) \nonumber \\
\label{eq:ind3}
 =& \!\!\!\!\!\!\sum_{(\Gamma,\ell, \bfp)\in {\rm RT}(g,\mu)_{1,2}} \!\!\!\!\! \frac{a(\Gamma,\ell,\bfp)}{|{\rm Aut}(\Gamma,\ell,\bfp)|} \,.
\end{align}
This formula together with Lemma~\ref{lem:inttree} thus 
implies Theorem~\ref{thm:indintall}. Since by definition
$\sum_{i=1}^n (m_i+1) = 2g-2 + n$, 
Lemma~\ref{lem:inttree} implies that
\begin{eqnarray*}
(2g-2+n) (m_1+1) a(\mu) &=& (m_1+1)^2   a(\mu)+  \sum_{i=2}^n (m_i+1)(m_1+1) a(\mu)\\
&=& (m_1+1)^2   a(\mu) + \sum_{i=2}^n \sum_{\begin{smallmatrix}  (\Gamma,\ell,\bfp) \in \\ {\rm RT}(g,\mu)_{1,i} \end{smallmatrix}} \frac{a(\Gamma,\ell,\bfp)}{|{\rm Aut}(\Gamma,\ell,\bfp)|}\\
&=&\sum_{\begin{smallmatrix}  (\Gamma,\ell,\bfp) \in \\ {\rm RT}(g,\mu)_{1}\end{smallmatrix}} \frac{a(\Gamma,\ell,\bfp)}{|{\rm Aut}(\Gamma,\ell,\bfp)|}\,.
\end{eqnarray*}
Therefore, the left-hand side of~\eqref{eq:ind3} can be rewritten as
\begin{eqnarray*}
&& \sum_{k \geq 1}\sum_{\bfg, \bfmu}
h_{\proj^1}((m_1,m_2),\bfp) \frac{1}{k!} \, \cdot\, 
\prod_{j=1}^k \! \sum_{\begin{smallmatrix}  (\Gamma,\ell,\bfp) \in \\
{\rm RT}(g_j,(p_j-1,\mu_j))_{1} \end{smallmatrix}}
\frac{a(\Gamma,\ell,\bfp)}{|{\rm Aut}(\Gamma,\ell,\bfp)|}\\
&=& \sum_{k \geq 1} \sum_{\bfg, \bfmu}
h_{\proj^1}((m_1,m_2),\bfp) \,\frac{1}{k!}\, \cdot\!\!
\sum_{\begin{smallmatrix}  (\Gamma_j,\ell_j,\bfp_j) \in \\
{\rm RT}(g_j,(p_j-1,\mu_j))_{1} \end{smallmatrix}}   \prod_{j=1}^k \frac{a(\Gamma_j,\ell_j,\bfp_j)}{|{\rm Aut}(\Gamma_j,\ell_j,\bfp_j)|}\,.
 \end{eqnarray*}
\par
We claim that there is a bijection
$$
{\rm RT}(g,\mu)_{1,2}\,\simeq\,  \bigcup_{(\Gamma',\ell',\bfp')\in \BB^\star_{1,2} }
\prod_{v\in V(\Gamma'),\ell(v)=0} {\rm RT}(g_v,(p_v-1,\mu_v))_{1}\,.
$$
Indeed given a rooted tree $(\Gamma,\ell,\bfp)$ in ${\rm RT}(g,\mu)_{1,2}$ we can
construct $(\Gamma',\ell',\bfp') \in \BB(g,n)^\star_{1,2}$ by contracting all edges
except those adjacent to the root, and the rooted trees $ (\Gamma_v,\ell_v,
\bfp_v)\in {\rm RT}(g_v,(p_v-1,\mu_v))_{1}$ are the connected components of the graph
obtained from $(\Gamma,\ell,\bfp)$ by deleting the root. Moreover for a rooted
tree $(\Gamma,\ell,\bfp)$, by equation~\eqref{eq:h(gamma)} we have 
$$
h(\Gamma_0, \ell_0, \bfp_0) = h_{\proj^1}((m_1,m_2), \bfp') \cdot \prod_{j=1}^k h(\Gamma_{j0}, \ell_{j0}, \bfp_{j0})\,,
$$
where as before $\Gamma_0$ is obtained from $\Gamma$ by removing the leaves and the $\Gamma_{j0}$ are the connected components after removing the root of $\Gamma_0$. Together with the definition of the $a$-contribution in~\eqref{eq:a-contribution}, it implies that 
$$
a(\Gamma,\ell, \bfp) = h_{\proj^1}((m_1, m_2), \bfp') \cdot \prod_{v\in V(\Gamma'),\ell(v)=0} a(\Gamma_v,\ell_v,\bfp_v)\,.
$$
Note also that 
$$
{\rm Aut}(\Gamma,\ell, \bfp) =  {\rm Aut}(\Gamma',\ell',\bfp')\times \prod_{v\in V(\Gamma'),\ell(v)=0}  {\rm Aut}(\Gamma_v,\ell_v, \bfp_v)\,.
$$
Combining the above we thus conclude that equality~\eqref{eq:ind3} holds.
\end{proof}
\par
\begin{proof}[Proof of the equivalence of Theorems~\ref{intro:IntFormula} and~\ref{intro:VolRec}]
We first assume that Theorem~\ref{intro:VolRec} holds. By Theorem~\ref{thm:indintall}, the
quantities 
\bes
{\rm vol}(\omoduli[g,n](m_1,\ldots,m_n)) \quad \text{and} \quad 
\frac{2(2\pi i)^{2g}}{(2g-3+n)!}\, a(m_1,\ldots,m_n)
\ees
satisfy the same induction relation that determines both collections of these 
numbers starting from the case $n=1$. The base case (i.e. the minimal strata) that 
\be \label{eq:volminviaa}
{\rm vol}(\omoduli[g,1](2g-2)) \=\frac{2(2\pi i)^{2g}}{(2g-2)!}\, a(2g-2)\,
\ee
was proved 
in~\cite{SauvagetMinimal} under a mild assumption of regularity of a 
natural Hermitian metric on $\mathcal{O}(-1)$, and we will give an alternative 
(unconditional) proof in Section~\ref{sec:appvol}. Consequently we conclude that 
Theorem~\ref{intro:VolRec} implies Theorem~\ref{intro:IntFormula}. 
The converse implication follows similarly. 
\end{proof}

