Let $D$ be the unit disk in the complex plane, and let $H$ be a fuchsian group generated by one fixed point free element, say $a$. What it the quotient $D/H$?
Attempt: The quotient is biholomorphic to the punctured disk if $a$ is parabolic (one fixed point on $\partial D$) and it is biholomorphic to an annulus if $a$ is hyperbolic (two fixed points on the boundary $\partial D$). I tried by considering the fundamental region of each of the fuchsian groups. In the parabolic case, for instance, the fundamental region is bounded by two geodesics both originating from the same point on $\partial \mathbb{D}$, and the action of $H$ gives the identification between the two geodesics, hence it is easy to see it is a punctured disk. Nevertheless, I could not come up with anything formal. Any help would be appreciated.
Remember that every parabolic transformation is conjuagate to the transformation $z\rightarrow z+1$ in the upper-half plane model, and that hyperbolic ones are conjugate to $z\rightarrow az$ for some $a$. This makes it a lot easier to choose exact fundamental domains.