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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.

Arun
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    Yes. This is usually known as Selberg's Lemma. Maybe a nice reference is https://www.jstor.org/stable/2373096?seq=1#metadata_info_tab_contents where it is stated as Lemma 2. – Grant Lakeland Jul 21 '19 at 22:31
  • Thank you, especially for the reference. – Arun Jul 23 '19 at 03:24

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