Clifford–Klein form

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.