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.

中文翻译：

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PSL2R$ {\mathrm{PSL}_2\mathbb{R}} $ 中纯双曲线表示的几何化

摘要 令 S 是 g 至少为 2 的曲面。如果表示 ρ:π1S→PSL2R $ \rho:\pi_1 S\to{\mathrm{PSL}_2\mathbb{R}} $ 被称为纯双曲线的，如果它的形象仅由双曲线元素和身份组成。我们可能想知道在什么条件下这样的表示会作为 S 上的分支双曲结构的完整而出现。在这项工作中，我们完全表征了它们，给出了充分必要条件。