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  "body": "\n## Statement\n\nLet $G$ be a connected [[semisimple Lie group|XJS07AJ50Y]] without compact factors, and let $\\Gamma$ be a [[lattice|BTYF5Y5PFY]] in $G$.\nLet $\\rho \\colon G \\to \\mathrm{GL}(V)$ be a continuous representation on a finite-dimensional real vector space $V$.\nThen:\n\n1. every linear subspace of $V$ that is [[invariant|0ACX5QT8TE]] under $\\rho(\\Gamma)$ is invariant under $\\rho(G)$;\n2. the closure of $\\rho(\\Gamma)$ in the [[Zariski topology|8SQK7RTB63]] of $\\mathrm{GL}(V)$ contains $\\rho(G)$.\n\n## Remarks\n\nFurstenberg's proof for clause 1 goes as follows.\nView a $\\Gamma$-invariant subspace $W$ of dimension $k$ as a point of the projective space $P$ of $\\Lambda^k V$.\nThe map $g\\Gamma \\mapsto \\rho(g) W$ pushes the invariant probability measure on $G/\\Gamma$ to a $G$-invariant probability measure on $P$.\nIf $\\rho(u)$ is unipotent, the points $\\rho(u)^m x$ converge to fixed points of $\\rho(u)$, so such a measure lives on these fixed points.\nAs $G$ is generated by such elements $u$, the measure lives on $G$-fixed points, and this forces $W$ to be $G$-invariant.\n\nExample: $\\Gamma = \\mathrm{SL}_2(\\Z)$ in $G = \\mathrm{SL}_2(\\R)$, with the standard representation on $\\R^2$.\nA line invariant under $\\begin{pmatrix} 1 \u0026 1 \\\\ 0 \u0026 1 \\end{pmatrix}$ must be $\\R e_1$, and a line invariant under $\\begin{pmatrix} 1 \u0026 0 \\\\ 1 \u0026 1 \\end{pmatrix}$ must be $\\R e_2$.\nSo no line is invariant under $\\Gamma$, as clause 1 predicts, since no line is invariant under $G$.\n\nThe hypothesis on compact factors is necessary.\nIn the compact group $G = \\mathrm{SO}(3)$, the trivial subgroup is a lattice, and every subspace of $\\R^3$ is invariant under it.\nBut the axis $\\R e_3$ is not invariant under $\\mathrm{SO}(3)$.\n\nConsequences: the centraliser of $\\Gamma$ in $G$ is the centre of $G$, and $\\Gamma$ is not contained in any proper algebraic subgroup.\n\n## Explanations\n\n### L1 human\n\nA lattice is a discrete subgroup with a quotient of finite volume, such as $\\mathrm{SL}_2(\\Z)$ in $\\mathrm{SL}_2(\\R)$.\nIt is a tiny, countable part of the big group.\nStill, the theorem says that no polynomial equation can tell the lattice apart from the big group.\nAny linear structure preserved by the lattice is preserved by the whole group.\n\n### L2 human\n\nThe Borel density theorem is the first step of many rigidity results.\nIt lets one pass from statements about $\\Gamma$ to statements about $G$, for instance in the [[Margulis superrigidity theorem|DSKV65NHZZ]], where Zariski density is part of the setup.\nIt also shows that the adjoint representation of $G$ restricted to $\\Gamma$ is irreducible when $G$ is simple, which is used for local rigidity of lattices.\n",
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