Example

Let $X$ be a complex analytic manifold, analytically isomorphic to a bounded open subset of $\mathbf{C}^n$, and let $G$ be the group of analytic automorphisms of $X$. The topology of compact convergence is compatible with the group structure of $G$, and it can be shown that $G$ operates properly on $X$. In particular, every discrete subgroup of $G$ operates properly on $X$.

Take for example $X$ to be the upper half-plane $\mathscr{J}(z) > 0$, which is analytically isomorphic to an open disc in $\mathbf{C}$. Then $G$ is the group of all transformations $z \to (az + b)/(cz + d)$, where $a, b, c, d$ are real and $ad - bc \neq 0$. The subgroup $H$ of $G$ which consists of all such transformations for which $a, b, c, d$ are integers and $ad - bc = 1$ is a discrete subgroup of $G$, called the modular group. By what has been said above, it operates properly on the upper half-plane $\mathscr{J}(z) > 0$.*