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  "body": "\n## Statement\n\nLet $G \\subseteq \\mathrm{GL}_n$ be a [[linear algebraic group|GA81VVQHEP]] defined over $\\Q$, and let $G^\\circ$ be its identity component.\nA *$\\Q$-character* of $G^\\circ$ is a homomorphism of algebraic groups $G^\\circ \\to \\mathrm{GL}_1$ defined over $\\Q$.\nLet $G(\\Z) = G(\\R) \\cap \\mathrm{GL}_n(\\Z)$, an [[arithmetic group|4XWG6S7R4Z]].\nThen $G(\\Z)$ is a [[lattice|NS2057MFWP]] in $G(\\R)$ if and only if $G^\\circ$ has no nontrivial $\\Q$-character.\n\nIn particular, if $G$ is semisimple, then $G(\\Z)$ is a lattice in $G(\\R)$.\n\n## Remarks\n\nThe proof uses reduction theory.\nOne builds an explicit Siegel set $\\mathfrak{S}$ in $G(\\R)$, of finite Haar measure, such that $G(\\R) = \\mathfrak{S} \\, F \\, G(\\Z)$ for a finite set $F$.\nFor $\\mathrm{GL}_n$ and $\\mathrm{SL}_n$, Siegel sets come from Minkowski's reduction of quadratic forms; the general case reduces to these.\n\nExample: $G = \\mathrm{SL}_n$ has no nontrivial characters, so $\\mathrm{SL}_n(\\Z)$ is a lattice in $\\mathrm{SL}_n(\\R)$.\n\nThe condition on characters is necessary.\nFor $G = \\mathrm{GL}_1$, the identity map is a $\\Q$-character.\nHere $G(\\Z) = \\set{\\pm 1}$, and $\\R^\\times / \\set{\\pm 1}$ has infinite Haar measure $\\int_0^\\infty dx / x$.\nSo $G(\\Z)$ is not a lattice in $G(\\R) = \\R^\\times$.\n\nA companion result, proved by Borel and Harish-Chandra and by Mostow and Tamagawa, says when $G(\\R)/G(\\Z)$ is compact.\nFor reductive $G$, this happens exactly when $G$ has no nontrivial unipotent elements in $G(\\Q)$ and no nontrivial $\\Q$-characters.\n\n## Explanations\n\n### L1 human\n\nTake a group of matrices defined by polynomial equations with rational coefficients, like the matrices of determinant $1$.\nIts integer points form a discrete subgroup of its real points.\nThe theorem says when this discrete subgroup is large enough that the quotient has finite volume.\nThe answer is: unless the group has a homomorphism to the multiplicative group, defined over $\\Q$, that would stretch the volume to infinity.\n\n### L2 human\n\nThis theorem is the source of most lattices in semisimple Lie groups.\nGroups like $\\mathrm{SL}_n(\\Z)$, $\\mathrm{Sp}_{2n}(\\Z)$ and integral orthogonal groups of indefinite forms are lattices because of it.\nThe [[Margulis arithmeticity theorem|FEJK846N5X]] gives a converse in higher rank: every irreducible lattice arises from this construction.\nFor $\\mathrm{SL}_2$ it recovers the fact that the modular surface $\\mathrm{SL}_2(\\Z) \\backslash \\mathbb{H}$ has finite area.\n",
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  "name": "Borel–Harish-Chandra theorem",
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