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Let  S  be a surface of genus  g  at least 2. A representationρ  :   π  1   S  →P  S  L   2    R$ \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.

Geometrisation of purely hyperbolic representations in P S L 2 R $ {\mathrm{PSL}_2\mathbb{R}} $