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  "body": "\n## Definition\n\nLet $G$ be a [[Lie group|7JBRGCVJRB]] and let $H$ be a closed subgroup of $G$.\nLet $X = G/H$ be the [[homogeneous space|85TWR6E75P]] of left cosets, with $G$ acting by left multiplication.\nLet $\\Gamma$ be a [[discrete subgroup|5M7D1BSHVZ]] of $G$ whose action on $X$ is [[properly discontinuous|J5X5ED0B2Y]].\nThe *Clifford–Klein form* of $X$ defined by $\\Gamma$ is the quotient space\n$$\\Gamma \\backslash X = \\Gamma \\backslash G / H,$$\nthe space of orbits of $\\Gamma$ on $X$.\nThe group $\\Gamma$ is then called a discontinuous group for $X$.\n\n## Remarks\n\nIn much of the literature $G$ is assumed to be reductive.\nIf $\\Gamma$ is torsion-free, its stabilizers, which are finite by proper discontinuity, are trivial, so $\\Gamma \\backslash X$ is a manifold locally modelled on $X$.\nExample: $G = \\mathrm{PSL}_2(\\R)$, $H = \\mathrm{PSO}(2)$, 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.\nHere $\\Gamma$ acts freely and properly discontinuously, and $\\Gamma \\backslash X$ is the closed hyperbolic surface.\nNon-example: $G = \\R^2$, $H = \\set{(t, \\alpha t) : t \\in \\R}$ with $\\alpha$ irrational, and $\\Gamma = \\Z^2$.\nThen $X = G/H \\cong \\R$ through $(x, y) \\mapsto y - \\alpha x$, and $(m, n) \\in \\Gamma$ acts by translation by $n - \\alpha m$.\nThese translations form a dense subgroup of $\\R$, so the orbits are not discrete and the action is not properly discontinuous.\n",
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