Hyperbolic cone-manifold structures with prescribed holonomy I: punctured tori

TL;DR

This paper characterizes when a representation of the punctured torus's fundamental group can be realized as a hyperbolic cone-manifold structure with a single corner point, linking algebraic properties to geometric structures.

Contribution

It provides a complete characterization for punctured tori, showing such representations are realizable if and only if they are not virtually abelian, and constructs explicit hyperbolic structures from these representations.

Findings

A representation is a holonomy of a hyperbolic cone-manifold with one corner point iff not virtually abelian.

Constructs a pentagonal fundamental domain from the representation's geometry.

Uses universal covering groups and Markoff moves to analyze the structures.

Abstract

We consider the relationship between hyperbolic cone-manifold structures on surfaces, and algebraic representations of the fundamental group into a group of isometries. A hyperbolic cone-manifold structure on a surface, with all interior cone angles being integer multiples of $2\pi$, determines a holonomy representation of the fundamental group. We ask, conversely, when a representation of the fundamental group is the holonomy of a hyperbolic cone-manifold structure. In this paper we prove results for the punctured torus; in the sequel, for higher genus surfaces. We show that a representation of the fundamental group of a punctured torus is a holonomy representation of a hyperbolic cone-manifold structure with no interior cone points and a single corner point if and only if it is not virtually abelian. We construct a pentagonal fundamental domain for hyperbolic structures, from the geometry of a representation. Our techniques involve the universal covering group of the group of orientation-preserving isometries of the hyperbolic plane, and Markoff moves arising from the action of the mapping class group on the character variety.