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I'm searching for a Fuchsian Group without fixed points. (because i need an example for a group $\Gamma$, so that $\mathbb{H}/\Gamma$ is a Riemannian surface, and therefore $\Gamma$ has to be a discrete subgroup of $PSL(2,\mathbb{R})$ which acts free and properly discontinous, it would be great if $\mathbb{H}/\Gamma$ is compact as well.) I found an example in "Svetlana Katok, Fuchsian Groups (Example C in Chapter 4)", but i think this one has fixed points...

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    I think page 142 of 'Fuchsian Groups' has a picture of a finite volume example.2018-06-25

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