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Sep 27, 2013 at 20:13 comment added Misha The standard definition of a hyperbolic Riemann surface is the one whose universal cover is biholomorphic to the unit disk. For instance, among compact Riemann surfaces this is equivalent to genus $>1$.
Sep 27, 2013 at 20:01 comment added Darius Math Ah, OK. I thought by hyperbolic Riemann surface you mean a Riemann surface with a number of points removed.
Sep 27, 2013 at 19:45 comment added Misha @DariusMath: In my examples $Y$ is compact.
Sep 27, 2013 at 18:29 comment added Darius Math Thanks a lot! In this example $Y$ is non-compact. Can it be that if we assume $Y$ to be a compact curve, there exists a homomorphism?
Sep 27, 2013 at 17:21 vote accept Darius Math
Sep 27, 2013 at 16:17 history answered Misha CC BY-SA 3.0