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  "body": "\n## Summary\n\nThe Zimmer program is the study of smooth actions of [[lattices|BTYF5Y5PFY]] in higher-rank [[semisimple Lie groups|XJS07AJ50Y]] on compact manifolds.\nIts guiding idea is that such actions are very restricted.\nIn low dimension they should be finite, and in general they should come from algebraic constructions.\nIt extends the [[Margulis superrigidity theorem|DSKV65NHZZ]], which describes linear representations of these lattices, to actions by diffeomorphisms.\n[[Zimmer's conjecture|JW71J3AKY4]] is its best known test case.\n\n## Explanations\n\n### L1 human\n\nA group such as $\\mathrm{SL}_3(\\Z)$ is huge and rigid.\nThe Zimmer program asks in which ways such a group can act by smooth symmetries on a compact space like a circle, a sphere or a torus.\nThe expected answer is that it can hardly act at all on small spaces, and that on larger spaces its actions come from known algebraic models.\nFor example, $\\mathrm{SL}_3(\\Z)$ acts on the torus $\\R^3/\\Z^3$ through its linear action on $\\R^3$, and actions of this kind are expected to be typical.\n",
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