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  "body": "\n## Statement\n\nLet $G$ be a connected [[semisimple Lie group|XJS07AJ50Y]] without compact factors.\nThen there is an open [[neighbourhood|0ZCY0Z80B6]] $U$ of the identity $e$ in $G$ with the following property.\nFor every [[discrete subgroup|5M7D1BSHVZ]] $\\Gamma$ of $G$ there is $g \\in G$ such that $g \\Gamma g^{-1} \\cap U = \\set{e}$.\n\n## Remarks\n\nThe proof starts from a [[Zassenhaus neighbourhood|FZ798EZ6M0]]: the elements of $\\Gamma$ close to $e$ generate a nilpotent group.\nOne then conjugates by suitable elements of $G$ that expand this nilpotent part, which is possible because $G$ has no compact factors.\n\nExample: $G = \\mathrm{SL}_2(\\R)$ and $\\Gamma = \\mathrm{SL}_2(\\Z)$.\nConjugating the element $\\begin{pmatrix} 1 \u0026 1 \\\\ 0 \u0026 1 \\end{pmatrix}$ by $\\operatorname{diag}(t, t^{-1})$ gives $\\begin{pmatrix} 1 \u0026 t^2 \\\\ 0 \u0026 1 \\end{pmatrix}$, which is close to $e$ for small $t$.\nSo some conjugates of $\\Gamma$ come close to $e$, and the theorem only asserts that some conjugate stays away.\n\nThe hypothesis on compact factors is necessary.\nIn the compact semisimple group $\\mathrm{SO}(3)$, let $\\Gamma_n$ be the finite cyclic group of rotations about one axis by multiples of $2\\pi/n$.\nEvery conjugate of $\\Gamma_n$ contains a rotation by the angle $2\\pi/n$.\nSo for a given $U$, every conjugate of $\\Gamma_n$ meets $U \\setminus \\set{e}$ once $n$ is large.\n\n## Explanations\n\n### L1 human\n\nA discrete subgroup has no elements arbitrarily close to the identity, but it might still have some fairly close ones.\nThe theorem says that, after moving the subgroup by a conjugation, one can keep all its elements outside a fixed neighbourhood of the identity.\nThe size of this neighbourhood depends only on the big group, not on the subgroup.\n\n### L2 human\n\nThe main consequence concerns [[lattices|BTYF5Y5PFY]].\nThe volume of $G/\\Gamma$ is at least a constant $c(G) \u003e 0$ that does not depend on the lattice $\\Gamma$.\nIndeed a small ball around $e$ injects into $G/(g\\Gamma g^{-1})$, and $G/\\Gamma$ and $G/(g\\Gamma g^{-1})$ have the same volume.\nFor quotients of symmetric spaces, this gives a uniform lower bound for the volume of locally symmetric orbifolds of a given type.\n",
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