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  "body": "\n## Statement\n\nLet $G$ be a connected [[semisimple Lie group|XJS07AJ50Y]] with finite center and without compact factors.\nAssume that no simple factor of $G$ is locally isomorphic to $\\mathrm{SL}_2(\\R)$ or to $\\mathrm{SL}_2(\\C)$.\nFix a [[Haar measure|VY0HXD3GC4]] on $G$, and for a [[lattice|BTYF5Y5PFY]] $\\Gamma$ of $G$ let $\\operatorname{covol}(\\Gamma)$ be the measure of $G/\\Gamma$ for the induced invariant measure.\nThen for every $v \u003e 0$ there are only finitely many lattices $\\Gamma$ in $G$ with $\\operatorname{covol}(\\Gamma) \\le v$, up to conjugation by elements of $G$.\n\n## Remarks\n\nThe result is due to H. C. Wang.\nThe proof combines two facts.\nThe [[Kazhdan–Margulis theorem|C34RZMKKA4]] gives a fixed neighbourhood of the identity that every lattice avoids up to conjugation, which makes the set of lattices of bounded covolume compact up to conjugation.\n[[Weil's local rigidity theorem|V2HTMG4KT1]] and its extension to non-uniform lattices make each lattice isolated up to conjugation, and a compact set of isolated points is finite.\nThe excluded factors are necessary.\nFix a genus $g \\ge 2$.\nIn $\\mathrm{PSL}_2(\\R)$ the fundamental groups of closed hyperbolic surfaces of genus $g$ all have covolume $4\\pi(g - 1)$, by the Gauss–Bonnet theorem.\nYet they form uncountably many conjugacy classes, parametrised by [[Teichmüller space|ZZGC6BZNJ5]].\nIn $\\mathrm{PSL}_2(\\C)$ hyperbolic Dehn filling gives infinitely many closed hyperbolic $3$-manifolds whose volumes are bounded by the volume of one cusped hyperbolic $3$-manifold.\n\n## Explanations\n\n### L1 human\n\nA lattice gives a finite-volume space $G/\\Gamma$, and one can measure its size.\nThe theorem says that, for most semisimple groups, only finitely many essentially different lattices have size below any given bound.\nSo lattices are sparse, unlike in the plane, where lattices can be deformed continuously.\nFor example, there are only finitely many closed hyperbolic manifolds of dimension at least $4$ with volume below any fixed number.\n\n### L2 human\n\nWang's theorem is the global consequence of two local facts: lattices cannot contain elements too close to the identity in a uniform way, and lattices cannot be deformed.\nIt fails for factors of type $\\mathrm{SL}_2(\\R)$, where lattices deform continuously.\nIt also fails for $\\mathrm{SL}_2(\\C)$, where hyperbolic Dehn filling produces infinitely many lattices of bounded covolume.\nIt is a starting point for counting lattices of bounded covolume, and for the study of the smallest covolume of a lattice in a given group.\n",
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