Hyperbolic 2-spheres with cone singularities

TL;DR

This paper investigates the geometric structure of hyperbolic 2-spheres with cone singularities, providing detailed descriptions for specific cases and linking to classical examples in complex hyperbolic geometry.

Contribution

It offers a detailed analysis of the moduli spaces of hyperbolic 2-spheres with cone points, especially for four points, connecting to Deligne-Mostow's nonarithmetic ball quotients.

Findings

Detailed description of hyperbolic 2-spheres with three cone points.

Identification of spaces related to Deligne-Mostow's examples.

Connections between hyperbolic cone spheres and complex hyperbolic geometry.

Abstract

We study the space $C(a_0,a_1,\dots,a_n)$ of hyperbolic 2-spheres with cone points of prescribed apex curvatures $2a_0,2a_1,\dots,2a_n\in]0,2\pi[$ and some related spaces. For $n=3$, we get a detailed description of such spaces. The euclidean 2-spheres were considered by W. P. Thurston: for $n=4$, the corresponding spaces provide the famous 7 examples of nonarithmetic compact holomorphic 2-ball quotients previously constructed by Deligne-Mostow.