Out of interest, I have been trying my hand at the following problem that is relevant to covering spaces:
Let $X$ be a locally compact and simply connected Riemann surface and let $G$ be a discrete subgroup of the group of automorphisms of $Aut(X)$ (here discrete is taken to mean that the identity element is an isolated point of $G$ within the Lie group $Aut(X)$). If $G$ acts freely on $X$, then the action of $G$ is properly discontinuous, that is, the set $$\{g \in G \ | \ K \cap g(K) \neq \emptyset\}$$ is finite for every compact subset $K \subset X$.
I have tried a sequence argument to no avail (unless I am missing something crucial) and I have tried searching the many questions relating to properly discontinuous actions that can be found on this site. However, I have yet to find an answer that is specific to this question so I decided to ask it myself. It would be greatly appreciated if I could be pointed in the right direction on this particular problem, or even directed to a highly similar question (specifically with this definition of properly discontinuous) that I have overlooked. Thanks!
Edit: It was made apparent to me in the comments that my original question might not end up being true, so I have imposed a further assumption that $X$ is a Riemann surface. If even now my question is not true, I would appreciate seeing either a counter-example or an explanation to how this is false. Thanks again!