# Penner's formula for volume of the Moduli Space

In his paper "Weil-Petersson Volumes" Penner gives the following formula for the integral of a top-dimensional cohomology class $\omega$ on the moduli space $\mathcal M_g^s$ of $s$-punctured riemann surfaces of genus $g$:

$$\int_{\mathcal M_g^s}\omega=\sum_G\frac{1}{|Aut(G)|}\int_{D(G)}\phi^*\omega$$

where the sum is taken over all isomorphism classes of ribbon graphs $G$, $\phi:\widetilde{\mathcal T_g^s}\to\mathcal M_g^s$ is the natural map from the decorated Teichmüller space to the moduli space, and $D(G)$ is more or less the pre-image under $\phi$ of the orbicell in $\mathcal M_g^s$ corresponding to $G$.

My question is to what extent can this formula be generalized. Specifically,

1) Is a similar formula for evaluating forms of arbitrary degree?

2) Is this formula the shadow'' of a general formula for evaluating cohomology classes on (smooth) orbifolds?

I myself cannot understand Penner's paper well enough to see where the top-dimensional assumption on $\omega$ enters the picture. References to the literature would be much appreciated as well.

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(of course, neither side makes sense if $\omega$ is not top dimensional) – John Pardon Jan 28 '12 at 18:56
Right. By similar formula of arbitrary degree'' I'm asking for the integral over a cycle of the appropriate dimension. – Steve Jan 29 '12 at 2:27
Isn't this formula essentially tautological, once you believe that this triangulation exists? i.e. a similar formula exists for any triangulated manifold (with no need for the 1/|Aut| factor for an honest triangulation, of course). – Tom Church Jan 29 '12 at 6:06

## 1 Answer

The major breakthrough in this was made by Maryam Mirzakhani in her thesis (see this preprint) Her stuff is is relevant to general intersection theory on moduli space (see references to Witten's and Weitsman's papers in her bibliography), which is one way to interpret your question about "not top-dimensional classes"). Her work has received a lot of attention (Google Scholar shows 83 citations) -- I am sure those 83 papers have a lot more information.

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The link appears to be broken. Do you know the title of the work it is supposed to link to? – Steve Jan 28 '12 at 21:51
– Aaron Bergman Jan 28 '12 at 22:11
Yes, sorry. Damn PC. – Igor Rivin Jan 28 '12 at 22:27
Should be fixed now. – Igor Rivin Jan 28 '12 at 22:30