Let $M$ be a compact hyperbolic Riemann surface. Is there a simple way to show that the automorphism group $Aut(M)$ of conformal self-mappings of $M$ is a finite group?
Recall that a hyperbolic Riemann surface is one whose universal cover is $\mathbb D$.
I was hoping there is some way to use basic covering space theory (especially the various Galois-like relations between groups and covers) and the fact the group of automorphisms of the unit disk is $PSL(2,\mathbb R)$, since we know $\mathbb D/ G\cong M$, where $G$ is the covering group of $M$. But it has been a while since I've studied covering space theory, and I'm not sure if this idea can be made to work. In particular, I'm not sure how to make use of the hypothesis that $M$ is compact.