It is known that (uniformization theorem) any Riemann surface can be written as the quotient of its universal cover by a discrete group (of Möbius transformations). This group is isomorphic to the fundamental group of the surface. My question is:
Can we choose a non-discrete group that is isomorphic to the fundamental group of the surface? What happens if we consider the quotient? (I know that it is not a surface in general).
In other words:
What happens if we quotient the upper-half plane by a non-discrete group?
Is it interesting to study such quotients? If so, why?