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  "body": "\n## Statement\n\nLet $G$ be a connected [[semisimple Lie group|XJS07AJ50Y]] with finite centre and no compact factors, of real rank at least $2$.\nLet $\\Gamma$ be an irreducible [[lattice|BTYF5Y5PFY]] in $G$: for every noncompact closed normal subgroup $N$ of $G$, the product $\\Gamma N$ is dense in $G$.\nThen $\\Gamma$ is arithmetic.\nThat is, there are a semisimple [[linear algebraic group|GA81VVQHEP]] $H \\subseteq \\mathrm{GL}_n$ over $\\Q$ and a continuous surjective homomorphism $\\phi \\colon H(\\R)^\\circ \\to G$ with compact kernel with the following property.\nThe groups $\\phi(H(\\Z) \\cap H(\\R)^\\circ)$ and $\\Gamma$ are commensurable: their intersection has finite [[index|0DTXBQPA2S]] in both.\n\n## Remarks\n\nThe proof rests on the [[Margulis superrigidity theorem|DSKV65NHZZ]].\nSuperrigidity is applied to $\\Gamma$ in real, complex and $p$-adic groups.\nIt shows that, in suitable coordinates, the matrix entries of $\\mathrm{Ad}(\\Gamma)$ lie in a number field and have bounded denominators.\nFrom this one builds the $\\Q$-group $H$.\n\nExample: $\\mathrm{SL}_n(\\R)$ with $n \\ge 3$ has real rank $n - 1 \\ge 2$.\nSo every lattice in it is commensurable, up to conjugation, with a group built from an algebraic group over $\\Q$, such as $\\mathrm{SL}_n(\\Z)$.\n\nThe rank condition cannot be dropped completely.\nGromov and Piatetski-Shapiro constructed non-arithmetic lattices in $\\mathrm{SO}(n, 1)$ for every $n \\ge 2$.\nMostow, and Deligne and Mostow, found non-arithmetic lattices in $\\mathrm{SU}(2, 1)$ and $\\mathrm{SU}(3, 1)$.\nIn real rank $1$, the groups $\\mathrm{Sp}(n, 1)$ and the rank one form of $F_4$ still have only arithmetic lattices, by Corlette and Gromov–Schoen.\n\n## Explanations\n\n### L1 human\n\nThe simplest way to build a lattice is to take the integer points of a group defined by equations, like $\\mathrm{SL}_3(\\Z)$ inside $\\mathrm{SL}_3(\\R)$.\nMargulis proved that in the big groups of higher rank, there is no other way.\nEvery lattice there is built from integer points, after allowing finite index changes and a few standard operations.\n\n### L2 human\n\nTogether with the [[Borel–Harish-Chandra theorem|3B451RD8AJ]], this gives a complete description of irreducible lattices in higher rank.\nIt turns questions about lattices into questions about algebraic groups over number fields.\nIn rank one, arithmeticity can fail, and hyperbolic manifolds give many non-arithmetic examples.\n",
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