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  "body": "\n## Definition\n\nLet $G$ be a connected [[semisimple Lie group|XJS07AJ50Y]] with finite center, all of whose simple factors have real rank at least $2$.\nLet $\\Gamma$ be a [[lattice|BTYF5Y5PFY]] in $G$.\nLet $M$ be a compact [[smooth manifold|F45VQ47GPV]] and $\\mathrm{Diff}(M)$ its group of [[diffeomorphisms|0KZZ70ZHXP]].\nThe *Zimmer program* is the research program that aims to classify the homomorphisms $\\rho \\colon \\Gamma \\to \\mathrm{Diff}(M)$, with these expected outcomes:\n\n1. If $\\dim M$ is small compared with the rank of $G$, then every such $\\rho$ has finite image.\n2. If $\\rho$ preserves a volume form, then, up to finite index, the action is built from isometric actions and from algebraic actions.\n   Here an algebraic action is an action on a quotient $\\Lambda \\backslash H$ of a Lie group $H$ by a lattice $\\Lambda$.\n   It comes from a homomorphism of $\\Gamma$ or $G$ into $H$.\n\n## Remarks\n\nExample of an action in the sense of clause 2: $\\mathrm{SL}_n(\\Z)$ acts on the torus $\\R^n / \\Z^n$ by $A \\cdot (x + \\Z^n) = Ax + \\Z^n$.\nThis is well defined because $A \\Z^n = \\Z^n$, and it preserves Lebesgue measure because $\\det A = 1$.\nClause 1 is sharp for $\\mathrm{SL}_n(\\Z)$: it acts with infinite image on the real projective space of dimension $n - 1$ through its linear action on lines of $\\R^n$.\nThe program is not a single theorem; clause 2 in particular has no fully agreed precise form.\nPrecise test cases are [[Zimmer's conjecture|JW71J3AKY4]] for clause 1 and conjectures on actions preserving a rigid geometric structure for clause 2.\n",
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