# Quotient under a Fuchsian Group

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.

• This post is about the hyperbolic case.
– 23rd
Commented Jun 13, 2013 at 20:08
• $a$ parabolic sounds like a limit rotation, in which case I don't see why th fundamental region should be a geodesic triangle. I imagine a strip between two limit-parallel geodesics. A hyperbolic transformation makes me think of a translation or glide reflection, in which case you'd have two fixed points on $\partial D$: the ideal endpoints of the axis of translation. The fundamental domain would be a strip delimited by hyper-parallel geodesics, and the orbifold formed from that using different identifications. Not sure I'll be able to help, but so far I feel like I'm missing something here.
– MvG
Commented Jun 14, 2013 at 8:35
• Edited. Now it makes more sense. Commented Jun 14, 2013 at 17:17

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.