---
tag: 27QRARKWD5
kind: concept
name: Kleinian group
msc: [30F40, 57K32, 20H10]
specializes: [5M7D1BSHVZ]
sources:
  - url: https://en.wikipedia.org/wiki/Kleinian_group
same_as:
  wikidata: Q4116484
  wikipedia: 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|GHVW1Z35BS]].
The theory links complex analysis, the geometry of $3$-manifolds, and fractal limit sets on the sphere.

## Explanations

### L1 human

Möbius transformations of the sphere, such as $z \mapsto (az + b)/(cz + d)$, move circles to circles.
A Kleinian group is a collection of these maps, closed under composition, with no maps arbitrarily close to the identity.
Repeating the maps of such a group often produces intricate fractal patterns of circles on the sphere.
The same maps act as rigid motions of three-dimensional hyperbolic space, and this is how they describe hyperbolic $3$-manifolds.