Recursion formula for the volumes of moduli spaces of compact hyperbolic surfaces with cone points

TL;DR

This paper derives a recursion formula for the Weil-Petersson volumes of moduli spaces of hyperbolic surfaces with cone points, generalizing Mirzakhani's results using McShane's identities.

Contribution

It introduces a recursion formula for volumes of moduli spaces with cone points, extending Mirzakhani's work to include surfaces with singularities.

Findings

Volumes are polynomial in boundary lengths and cone angles.

Recursion formula generalizes Mirzakhani's volume computations.

Uses generalized McShane's identities for derivation.

Abstract

Let $V_{g,m,n}(\overrightarrow L,\overrightarrow \theta)$ be the Weil-Petersson volume of the moduli space of hyperbolic surfaces of genus g with m geodesic boundary components of length $\overrightarrow L=(\ell_1,...,\ell_m)$ and $n$ cone points of angle $\overrightarrow \theta=(\theta_1,...\theta_n)$. By using the generalized McShane's identities, we show that $V_{g,m,n}(\overrightarrow L,\overrightarrow \theta)$ is a polynomial of $(\ell_1,...,\ell_n,i\theta_1,...,i\theta_m)$. And we obtain a recursion formula for $V_{g,m,n}(\overrightarrow L,\overrightarrow \theta)$, which is a generalization of Mirzakhani's result.