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I was told relatively vaguely that for some discrete subgroup $\Gamma$ of $SL(2, \mathbb{R})$ (not necessarily contained in $SL(2,\mathbb{Q})$) and a subset $U$ of $\mathbb{C}$, the quotient $\Gamma\backslash U$ can be made into a Riemann surface if $\Gamma$ acts on $U$. Adding cusps with infinitely many stabilizers to the quotient compactifies the surface.

I was wondering if there are some texts/papers that I can study to get a general picture of this process. Obviously for $SL(2, \mathbb Z)$ one could follow the analytic construction with classical modular forms. But, as a person who's not very familiar with algebraic curves and has never heard of the statement of the previous "theorem", I am eager to read some more general and relevant materials.

Thanks in advance. Any help will be appreciated.

GoogleME
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For instance, if $\Gamma$ is a discrete subgroup of $PSL(2, {\mathbb C})$ and $U$ is an open subset of the extended complex plane on which $\Gamma$ acts properly discontinuously, then $S=U/\Gamma$ always has a natural structure of a Riemann surface, i.e. such that the quotient map $U\to S$ is holomorphic. See for instance, Theorem 6.2.1 in

Beardon, Alan F., The geometry of discrete groups, Graduate Texts in Mathematics, 91. New York - Heidelberg - Berlin: Springer-Verlag. XII, 337 p. (1983). ZBL0528.30001.

For instance, this works if $\Gamma< PSL(2, {\mathbb R})$ and $U$ is the upper half-plane.

In general, there is no way to compactify this Riemann surface by adding countably many points. However, if $U/\Gamma$ has finite hyperbolic area, then yes, adding cusps you can compactify it. See again Beardon's book.

As a general suggestion, if somebody tells you something vague, ask for references.

Moishe Kohan
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