# Fuchsian Group without fixed points

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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## 1 Answer

You may be thinking too hard about this. For example, the group generated by $z\mapsto z+1$ (where I am considering $z\in\mathbb{H}$, the upper half plane), is discrete, fixed point free, and acts properly discontinuously on $\mathbb{H}$; the quotient is a topological cylinder.

Actually writing down generators a surface group in $SL_2\mathbb{R}$ is usually unilluminating. Often, one instead regards a closed surface $S$ as obtained by gluing a polygon, and then presents the polygon in $\mathbb{H}^2$ with a right angle at each vertex. Then Poincare's polygon theorem gives a presentation of the surface group and $\mathbb{H}$ modulo this group is homeomorphic to $S$.

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