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  "body": "\n## Statement\n\nLet $n \\ge 3$ and let $\\Gamma$ be a [[lattice|BTYF5Y5PFY]] in the [[special linear group|0V9PMN0RBH]] $\\mathrm{SL}_n(\\R)$, for example $\\Gamma = \\mathrm{SL}_n(\\Z)$.\nLet $M$ be a compact [[smooth manifold|F45VQ47GPV]] with $\\dim M \u003c n - 1$, and let $\\mathrm{Diff}(M)$ be its group of [[diffeomorphisms|0KZZ70ZHXP]].\nThen every group homomorphism $\\rho \\colon \\Gamma \\to \\mathrm{Diff}(M)$ has finite image.\n\n## Remarks\n\nThis is the case of $\\mathrm{SL}_n(\\R)$ of a more general conjecture of Zimmer for lattices in higher-rank simple Lie groups, with a dimension bound depending on the group.\nZimmer stated conjectures of this kind in his 1986 address to the International Congress of Mathematicians, in several versions, some of them only for actions preserving a volume form.\nThe bound $n - 1$ is sharp: $\\mathrm{SL}_n(\\Z)$ acts with infinite image on the real projective space of dimension $n - 1$.\nAaron Brown, David Fisher and Sebastián Hurtado proved it, with $C^2$ regularity, for cocompact lattices in $\\mathrm{SL}_n(\\R)$.\nIn a paper published in Inventiones Mathematicae in 2020 they proved it for actions by $C^2$ diffeomorphisms of finite-index subgroups of $\\mathrm{SL}_n(\\Z)$ with $n \u003e 3$.\nThe general conjecture for lattices in other higher-rank groups is still studied.\n\n## Explanations\n\n### L1 human\n\nThe group $\\mathrm{SL}_n(\\Z)$ of integer matrices with determinant $1$ is a very rigid infinite group when $n \\ge 3$.\nThe conjecture says it cannot act in an interesting way by smooth symmetries on a compact space whose dimension is too small.\nAny such action must factor through a finite group.\nSo a big rigid group needs room to move, and a low-dimensional manifold does not give it enough.\n",
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      "first_stated": "1986",
      "posed_by": "Robert Zimmer"
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    "kind": "conjecture",
    "msc": [
      "37C85",
      "57S20",
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