Kleinian group

Summary

A Kleinian group is a discrete subgroup of $\mathrm{PSL}_2(\C)$. The group $\mathrm{PSL}_2(\C)$ acts on the Riemann sphere by Möbius transformations and on hyperbolic $3$-space by orientation-preserving isometries. Quotients of hyperbolic $3$-space by torsion-free Kleinian groups are exactly the complete oriented hyperbolic 3-manifolds. The theory links complex analysis, the geometry of $3$-manifolds, and fractal limit sets on the sphere.