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

Let $S$ be a surface of genus $g$ at least $2$. A representation $\rho :{\pi }_{1}S⟶{\text{PSL}}_2\mathbb{R}$ 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.