Suppose I have a bounded, connected Riemann surface whose automorphism group is infinite. Does it follow (and if yes why) that the Riemann surface is homeomorphic to a disk or an annulus?
Trying to find something about this question, I have found Hurwitz's Automorphism Theorem, from which it seems to follow that every compact Riemann surface with more than one hole must have finite automorphism group. Unfortunately I am no expert in geometry, so I am not sure if I can apply this theorem here. In particular, my Riemann surface is not compact but open, so this might be a problem. Or anyway, is there an easier way to prove the claim?