Volumes of moduli spaces of hyperbolic surfaces with cone points

TL;DR

This paper investigates the volumes of moduli spaces of hyperbolic surfaces with various boundary types, computes new cases involving large cone angles, and explores the geometric implications of Mirzakhani's polynomials with imaginary boundary lengths.

Contribution

It introduces new volume computations for hyperbolic surfaces with cone points and interprets Mirzakhani's polynomials in the context of hyperbolic cone angles.

Findings

Computed volumes for cases with large cone angles

Provided geometric interpretation of Mirzakhani's polynomials with imaginary boundary lengths

Analyzed volume behavior as cone angles approach 2π

Abstract

In this paper we study volumes of moduli spaces of hyperbolic surfaces with geodesic, cusp and cone boundary components. We compute the volumes in some new cases, in particular when there exists a large cone angle. This allows us to give geometric meaning to Mirzakhani's polynomials under substitution of imaginary valued boundary lengths, corresponding to hyperbolic cone angles, and to study the behaviour of the volume under the $2\pi$ limit of a cone angle.