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?