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  "body": "\n## Statement\n\nLet $G$ be a connected [[semisimple Lie group|XJS07AJ50Y]] with finite center.\nAssume that $G$ has no compact factor and no factor locally isomorphic to $\\mathrm{SL}_2(\\R)$.\nLet $\\Gamma$ be a uniform [[lattice|BTYF5Y5PFY]] in $G$, so that $G/\\Gamma$ is compact, and let $\\iota \\colon \\Gamma \\to G$ be the inclusion.\nGive the set $\\Hom(\\Gamma, G)$ of [[group homomorphisms|0JYTG5K9Y4]] the topology of pointwise convergence.\nThen $\\iota$ is locally rigid: $\\iota$ has a neighbourhood $U$ in $\\Hom(\\Gamma, G)$ with the following property.\nFor every $\\rho \\in U$ there is $g \\in G$ with $\\rho(\\gamma) = g \\gamma g^{-1}$ for all $\\gamma \\in \\Gamma$.\n\n## Remarks\n\nA uniform lattice is finitely generated, so the topology on $\\Hom(\\Gamma, G)$ is that of a real algebraic set, given by the images of finitely many generators.\nWeil proved a general criterion for a homomorphism $\\rho$ from a finitely generated group $\\Gamma$ to $G$.\nLet $\\Gamma$ act on the Lie algebra $\\mathfrak{g}$ of $G$ through $\\rho$ and the adjoint representation.\nIf the first cohomology group $H^1(\\Gamma, \\mathfrak{g})$ vanishes, then $\\rho$ is locally rigid.\nThe tangent space to deformations of $\\rho$ is the space of cocycles, and conjugations give the coboundaries, so vanishing of $H^1$ means that every infinitesimal deformation is a conjugation.\nHe then proved this vanishing for uniform lattices under the hypotheses above.\nThe hypothesis on $\\mathrm{SL}_2(\\R)$ is necessary.\nThe fundamental group of a closed hyperbolic surface sits in $\\mathrm{PSL}_2(\\R)$ as a uniform lattice, and it has a continuous family of non-conjugate deformations, described by [[Teichmüller space|ZZGC6BZNJ5]].\nGarland and Raghunathan extended the result to non-uniform lattices in groups of real rank one other than those locally isomorphic to $\\mathrm{SL}_2(\\R)$ and $\\mathrm{SL}_2(\\C)$.\n\n## Explanations\n\n### L1 human\n\nA uniform lattice is a discrete group of symmetries that tiles a big continuous group with compact tiles.\nThe theorem says that one cannot wiggle such a lattice slightly to get a genuinely new one.\nEvery small change of the lattice is just the same lattice seen from a different point, that is, conjugated by an element of $G$.\nThe only exception is the group $\\mathrm{SL}_2(\\R)$ and its relatives, where surfaces can be deformed.\n\n### L2 human\n\nLocal rigidity is the first of a chain of rigidity results for lattices.\n[[Mostow rigidity|4BEWKR0EG7]] upgrades it to a global statement: an abstract isomorphism between lattices extends to the ambient groups.\n[[Margulis superrigidity|DSKV65NHZZ]] goes further for higher rank, by controlling all homomorphisms of a lattice into other groups.\n",
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    "name": "Weil's local rigidity theorem",
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  "msc": [
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  "name": "Weil's local rigidity theorem",
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