By the uniformization theorem, any Riemann surface $H$ is conformally equivalent to one of constant curvature. Let's say that $H$ is conformally equivalent to a surface with curvature $-1$.
Therefore, the hyperbolic plane is a conformal universal cover of $H$. For any tiling on $H$, there is a corresponding tiling in the hyperbolic plane.
My question is, how do you figure out for which tilings in the hyperbolic plane there exists a tiling in $H$ that corresponds to it.
EDIT: For example, perhaps your surface is made by sewing together polygons, and the tiling in question is a {m,n} tiling. (This in particular makes the problem a computational decision problem.)