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Let $S$ be a compact Riemann surface with genus $\geq$ 2. Then by the Uniformization theorem it has universal covering space the upper half-plane $\mathbb{H}$(up to conformal equivalence). Now I read that we can express $S$ as $\mathbb{H}/\Gamma$, where $\Gamma$ is a stricly hyperbolic Fuchsian group. The statement sounds reasonable, but I want to have a actual proof of that. Can someone suggest a reference for that? Thanks a lot.

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    $\begingroup$ This is proved in Benedetti and Petronio's book Lectures on Hyperbolic Geometry in proposition B.1.6. $\endgroup$ – Camilo Arosemena-Serrato Apr 9 at 16:29
  • $\begingroup$ @CamiloArosemena-Serrato That one helps a lot. Thank you very much. $\endgroup$ – scd Apr 10 at 3:06

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