I just want to confirm if I am right here. Can I say every Fuchsian group corresponds to a way to glue the hyperbolic plane to get a suface?
Then the Poincare polygon theorem means that, given a convex finitely sided polygon and side pairing with appropriate angle sums of vertex cycle, we can find a Fuchsian group such that this polygon is fundamental (and thus tessellate the corresponding surface).
I am not sure if this is correct because my intuition here is on complex plane (rather than on hyperbolic plane), where the Fuchsian group is $\langle +1, -1, +i, -i\rangle$ and the fundamental polygon can be the unit square. Then the Fuchsian group corresponds to a torus and the unit square forms a tessellation.