Geometrization of purely hyperbolic representations in $\text{PSL}_2\Bbb R$

TL;DR

This paper characterizes when purely hyperbolic representations of surface groups into PSL(2,R) can be realized as holonomies of hyperbolic cone-structures on the surface, providing complete necessary and sufficient conditions.

Contribution

It offers a complete characterization of purely hyperbolic representations as holonomies of hyperbolic cone-structures, filling a gap in understanding their geometric realization.

Findings

Provides necessary and sufficient conditions for such representations to be holonomies.

Characterizes the geometric structures associated with purely hyperbolic representations.

Advances the understanding of the relationship between algebraic representations and geometric structures.

Abstract

Let $S$ be a surface of genus $g$ at least $2$. A representation $\rho:\pi_1S\longrightarrow \text{PSL}_2\Bbb R$ is said to be purely hyperbolic if its image consists only of hyperbolic elements other than the identity. We may wonder under which conditions such representations arise as holonomy of a hyperbolic cone-structure on $S$. In this work we will characterize them completely, giving necessary and sufficient conditions.