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

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.