Large cone angles on a punctured sphere

TL;DR

This paper investigates the moduli space of punctured spheres with large cone angles up to 2π, extending previous results valid for angles less than π, and provides explicit geometric and symplectic descriptions.

Contribution

It introduces a coordinate system related to Penner's λ-lengths for cone angles up to 2π and analyzes the mapping class group action and Wolpert's symplectic form in this setting.

Findings

Explicit coordinate system for cone angles up to 2π

Calculation of the mapping class group action in these coordinates

Expression of Wolpert's symplectic form in the new coordinates

Abstract

Do and Norbury found a so-called differential relation which relates the volume of the moduli space of singular surface with a cone point to that of a smooth surface obtained by forgetting the cone point. Their procedure is valid for cone angles less than $\pi$ by work of Tan, Wong and Zhang. We study the moduli space of a surface with a single cone point of angle ranging from $0$ to $2\pi$ using a coordinate system closely related to Penner's $\lambda$-lengths. We compute the action of the mapping class group in these coordinates, give an explicit expression for Wolpert's symplectic form and use this to justify Do and Norbury's approach using just hyperbolic geometry.