Bianchi Groups (reviewed)

Let $K$ be an imaginary quadratic field and $\mathcal{O}$ be its ring of integers. Then the group $$\textrm{PSL}_2(\mathcal{O})$$ is called the Bianchi group associated to $K$.

Bianchi groups are discrete subgroups of $\textrm{PSL}_2(\mathbb{C})$. The latter can be identified with the group of orientation-preserving isometries of the hyperbolic 3-space $\mathcal{H}_3$ and as a result, every Bianchi group $G$ acts properly discontinuously on $\mathcal{H}_3$. The quotient space $G \backslash \mathcal{H}_3$ is a non-compact hyperbolic 3-fold with finite volume.