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Which surfaces arise as quotients $\mathbb{H}^2/\Gamma$ where $\Gamma$ is a discrete subgroup of $PSL_2(\mathbb{R})$ which acts freely on $\mathbb{H}^2$?

The uniformization theorem tells us that any hyperbolic structure on a compact surface $S$ (where the induced metric is complete) is of the form $S= \mathbb{H}^2/\Gamma$ for some discrete subgroup $\Gamma$ of $PSL_2(\mathbb{R})$ which acts freely on $\mathbb{H}^2$.

I wanted to know in some sense what the converse is. Which surfaces arise when we look at $S= \mathbb{H}^2/\Gamma$ for an arbitrary discrete freely acting subgroup? Can we get non-compact surfaces (or more specifically can we get surfaces with cusps)? Can we get surfaces with boundary?

I was thinking about how to get $\mathbb{H}^2/\Gamma$ to be an ideal triangle. Although I think that $\Gamma$ would then have to be generated by reflections in the sides of an ideal triangle in $\mathbb{H}^2$, and would therefore not act freely?

Joey Brew
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    The only connected surfaces which you cannot get this way are the sphere, projective plane, torus, and Klein bottle. Every other surface (including every one which has infinitely generated fundamental group) arises in this way. – Lee Mosher May 28 '18 at 03:56
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    But, surfaces with boundary cannot be obtained as quotients of free actions on $\mathbb{H}^2$ itself, because $\mathbb{H}^2$ has empty boundary. – Lee Mosher May 28 '18 at 03:58

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