Quotient of Riemann surface by properly discontinuous group action I am looking for a reference on quotients of Riemann surfaces by properly discontinuous group actions.
In particular, I would like a reference for the following result: Let $M$ be a Riemann surface and let $G$ be a subgroup of $\text{Aut}(M)$ whose action on $M$ is properly discontinuous. Then the quotient space $M/G$ has the structure of a Riemann surface, and the projection map $\pi\colon M\to M/G$ is holomorphic.
Edit: My definition of "properly discontinuous" is that for any compact subset $K$ of $M$, the set $\{g\in G:K\cap g(K)\neq\varnothing\}$ is finite.

For the first part, you might want to take a look at Theorem 6.2.1. in Beardon's Geometry of Discrete Groups:
Theorem 6.2.1. Let $D$ be a subdomain of $\hat{\mathbb{C}}$ and let $G$ be a group of Möbius transformations which leaves $D$ invariant and which acts discontinuously in $D$. Then $D/G$ is a Riemann surface.
Beardon gives a detailed proof over several pages.
Chapter III Section 3 of Miranda - Algebraic Curves and Riemann Surfaces might also be interesting as well as Theorem 5.9.1 in Jones & Singerman - Complex Functions. Here the second part is also shown.
Non of these sources contain the exact statement you are looking for, but may contain ingredients for a proof.
