Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 24 / 5 / a

A Kleinian group is a discrete subgroup of the orientation-preserving isometry group of ${\mathbb{H}}^3$, identified with ${\mathrm{PSL}}_2(\mathbb{C})$. For an abstract finitely generated group this means a faithful representation into ${\mathrm{PSL}}_2(\mathbb{C})$ whose image is discrete; the chosen image is the geometric realization. Its boundary action is by Möbius transformations on $S_{\infty }^2\cong \mathbb{C}\cup \{\infty \}$.

Discreteness means the identity is isolated in the group topology, and for a subgroup it also excludes an infinite sequence of distinct elements converging anywhere in the isometry group. Torsion-freeness is not part of this definition: elliptic Möbius transformations of finite order are allowed. Therefore a Kleinian quotient need not be a smooth hyperbolic manifold.