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  "body": "\n## Statement\n\nLet $G$ be a connected [[Lie group|7JBRGCVJRB]], and let $\\Gamma$ be a [[lattice|BTYF5Y5PFY]] in $G$.\nLet $H$ be a connected closed subgroup of $G$ generated by one-parameter subgroups $(u_t)$ such that each $\\mathrm{Ad}(u_t)$ is unipotent.\nThen for every $x \\in G/\\Gamma$ there is a closed subgroup $L$ of $G$ containing $H$ with the following property.\nThe [[closure|0QEAJB91GZ]] of the [[orbit|0YQPZB5Q1G]] $Hx$ equals the orbit $Lx$, and $Lx$ carries an $L$-invariant probability measure.\n\n## Remarks\n\nThe statement was conjectured by M. S. Raghunathan and proved by Marina Ratner in 1991.\nHer proof derives it from her classification of the invariant measures of unipotent flows, one of [[Ratner's theorems|3MXR03096V]].\nBefore that, Margulis studied $H = \\mathrm{SO}(2, 1)^\\circ$ acting on $\\mathrm{SL}_3(\\R)/\\mathrm{SL}_3(\\Z)$.\nHe proved that every relatively compact $H$-orbit there is closed, and deduced the Oppenheim conjecture.\n\nExample: for $G = \\R^2$, $\\Gamma = \\Z^2$ and $H$ a line through $0$ of slope $\\alpha$, all elements are unipotent.\nIf $\\alpha$ is rational, each orbit of $H$ on the torus is a closed circle, and $L = H$.\nIf $\\alpha$ is irrational, each orbit is dense, and $L = G$.\n\nThe unipotent hypothesis is necessary: orbits of the diagonal group in $\\mathrm{SL}_2(\\R)/\\mathrm{SL}_2(\\Z)$ can have closures that are not orbits of a subgroup.\n\n## Explanations\n\n### L1 human\n\nLines of rational slope on a torus close up, and lines of irrational slope fill the torus.\nThe conjecture says that the same simple dichotomy, in a more general form, holds for orbits of unipotent groups on all spaces of the form $G/\\Gamma$.\nThe closure of an orbit is never a messy set: it is always a smooth homogeneous piece of the space.\n\n### L2 human\n\nThe theorem is the topological shadow of Ratner's measure classification.\nIt is the main tool for questions about the values of polynomials at integer points.\nThe first such question was the Oppenheim conjecture.\nLet $Q$ be a nondegenerate indefinite real quadratic form in $n \\ge 3$ variables, not a multiple of a form with rational coefficients.\nThen the values $Q(\\Z^n)$ are dense in $\\R$.\nThe theorem shows that orbit closures of unipotent flows behave like closures of linear flows on tori, but on noncommutative spaces.\n",
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  "msc": [
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  "name": "Raghunathan's conjecture",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/ME4/ME4ZSRAT5Z.md",
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