---
tag: V0SGQ7E092
kind: definition
name: Clifford–Klein form
msc: ["57S30", "22E40", "22F30"]
defines: R73WRFDXRY
uses: [7JBRGCVJRB, 85TWR6E75P, 5M7D1BSHVZ, J5X5ED0B2Y]
sources:
  - url: https://en.wikipedia.org/wiki/Clifford–Klein_form
---

## Definition

Let $G$ be a [[Lie group|7JBRGCVJRB]] and let $H$ be a closed subgroup of $G$.
Let $X = G/H$ be the [[homogeneous space|85TWR6E75P]] of left cosets, with $G$ acting by left multiplication.
Let $\Gamma$ be a [[discrete subgroup|5M7D1BSHVZ]] of $G$ whose action on $X$ is [[properly discontinuous|J5X5ED0B2Y]].
The *Clifford–Klein form* of $X$ defined by $\Gamma$ is the quotient space
$$\Gamma \backslash X = \Gamma \backslash G / H,$$
the space of orbits of $\Gamma$ on $X$.
The group $\Gamma$ is then called a discontinuous group for $X$.

## Remarks

In much of the literature $G$ is assumed to be reductive.
If $\Gamma$ is torsion-free, its stabilizers, which are finite by proper discontinuity, are trivial, so $\Gamma \backslash X$ is a manifold locally modelled on $X$.
Example: $G = \mathrm{PSL}_2(\R)$, $H = \mathrm{PSO}(2)$, so that $X$ is the hyperbolic plane, and $\Gamma$ the fundamental group of a closed surface of genus $g \ge 2$ embedded as a cocompact discrete subgroup.
Here $\Gamma$ acts freely and properly discontinuously, and $\Gamma \backslash X$ is the closed hyperbolic surface.
Non-example: $G = \R^2$, $H = \set{(t, \alpha t) : t \in \R}$ with $\alpha$ irrational, and $\Gamma = \Z^2$.
Then $X = G/H \cong \R$ through $(x, y) \mapsto y - \alpha x$, and $(m, n) \in \Gamma$ acts by translation by $n - \alpha m$.
These translations form a dense subgroup of $\R$, so the orbits are not discrete and the action is not properly discontinuous.