Prove that a group of Möbius transformations is discrete if, and only if, it acts discontinuously on hyperbolic 3 -space.

Let $G$ be the set of Möbius transformations $z \mapsto \frac{a z+b}{c z+d}$ with

$$a, b, c, d \in \mathbb{Z}[i]=\{u+i v: u, v \in \mathbb{Z}\} \quad \text { and } \quad a d-b c=1$$

Show that $G$ is a group and that it acts discontinuously on hyperbolic 3-space. Show that $G$ contains transformations that are elliptic, parabolic, hyperbolic and loxodromic.