Examples of quotients by Kleinian groups

Question

Let $G$ be a subgroup of $PSL(2,\mathbb{C})$, so that $G$ acts on $\mathbb{C}\cup\{\infty\}$ by linear fractional transformations. We say that $G$ acts properly discontinuously at a point $z\in \mathbb{C}\cup\{\infty\}$ if

1. the stabilizer $G_z$ is finite, and
2. there exists a neighborhood $U_z$ such that $g(U_z)=U_z$ for any $g$ in $G_z$ and $U_z\cap g(U_z)=\emptyset$ for any $g$ in $G\setminus G_z$.

Let $\Omega(G)$ be the subset of $\mathbb{C}\cup\{\infty\}$ at every point of which $G$ acts properly discontinuously. It is an open and $G$-invariant subset of $\mathbb{C}\cup\{\infty\}$. There is a theorem (see e.g. 1.5.2.5.1 Theorem from here) saying that $\Omega(G)/G$ can be endowed with a complex structure so that it becomes a Riemann surface provided that $\Omega(G)\ne \emptyset$ (and say $\Omega(G)$ is connected).

I was wondering what is the simplest non-trivial example of an application of this theorem? Of course one may take the disc and a finite group of its rotations about the origin, and the quotient will again be the disc.

• One may also take the upper half-plane and take its quotient by $PSL(2,\mathbb{Z})$, but how to describe the quotient?

• Also, are there simpler examples than $\mathbb{H}/PSL(2,\mathbb{Z})$ (when the group is infinite)?

Answer 1

I will start off with simpler examples. You can take an infinite discrete cyclic subgroup, and in the quotient you get an annulus (with infinite area). Moreover any surface which is hyperbolic can be written an $\mathbb{H}/\Gamma$ with gamma a discrete subgroup of $PSL(2,\mathbb{R})$. For example, genus $g$ surfaces with $g>1$. The typical way to describe the $\Gamma$ is through fundamental domain of the action, or construct fundamental domains that behave nicely. Fuchsian Groups by Svetlana Katok actually goes through the details, and writes an explicit $\Gamma$ in terms of matrices for surface of genus $2$, and describes the process, which is using fundamental domains and some hyperbolic geometry.

The typical fundamental domain given to $\mathbb{H}/PSL(2,\mathbb{Z})$ is a triangle with one ideal vertex, equal to $\{z \mid |z| \geq 1, |\Re (z)| \leq 1/2 \}$. The quotient is an orbifold (since the group action has fixed points) with a cone points of order $3$ and one of order $2$. A way to visualize the space is to fold the triangle across the order 2 point $i$, which will give a "long triangular pillow". Topologically the space is a punctured sphere/plane.