# Reference request: Fuchsian Model of compact Riemann surface

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.

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