Suppose I have a polygon $F$ and a group $G$ of isometries of the upper half plane, $\mathbb{H}^2$, such that $F$ tessellate $\mathbb{H}^2$ under the action of $G$. Can we always say that $G$ has a finite index discrete subgroup which acts freely on the upper half plane?
This is true if $G$ is a triangle group. But I am not sure if it is true in general.