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.
Every complete Riemannian 2-manifold of constant curvature $-1$ has $H^2$ as a universal cover, and its deck transformation group acts freely and discretely on $H^2$ and is thus a Fuchsian group, but not every Fuchsian group acts freely, so not every quotient of $H^2$ by a Fuchsian group is a Riemannian manifold. For example, if $r$ is a point reflection across some point $p \in H^2$, then $G := \{1, r\}$ is a Fuchsian group, but $H^2/G$ has a cone point at $p$.