Geometrisation of purely hyperbolic representations in PSL2R $ {\mathrm{PSL}_2\mathbb{R}} $

Abstract

Abstract Let S be a surface of genus g at least 2. A representation ρ:π1S→PSL2R $ \rho:\pi_1 S\to{\mathrm{PSL}_2\mathbb{R}} $ is said to be purely hyperbolic if its image consists only of hyperbolic elements along with the identity. We may wonder under which conditions such representations arise as the holonomy of a branched hyperbolic structure on S. In this work we characterise them completely, giving necessary and sufficient conditions.