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  "body": "\n## Summary\n\nA locally compact group has property (T), or Kazhdan's property (T), under the following condition.\nEvery unitary representation that almost has invariant vectors actually has a nonzero invariant vector.\nKazhdan introduced it to show that lattices in groups such as $\\mathrm{SL}_n(\\R)$ with $n \\ge 3$ are finitely generated.\nIt is a strong rigidity property, opposite to [[amenability|CEAT9BESKK]]: a locally compact group that is both amenable and has property (T) is compact.\nIt has applications to expander graphs, to ergodic theory and to the theory of operator algebras.\n\n## Explanations\n\n### L1 human\n\nLet a group act by rotations, or more generally by unitary maps, on a space of possibly infinite dimension.\nSuppose some unit vector is moved only a tiny bit by each element of a fixed generating set.\nFor most groups this tells you nothing, but for groups with property (T) it forces the existence of a vector that is not moved at all.\nThe integers fail this: shifting a long flat block of ones in $\\ell^2(\\Z)$ moves it very little, yet no nonzero vector of $\\ell^2(\\Z)$ is fixed by all shifts.\nThe groups $\\mathrm{SL}_n(\\Z)$ with $n \\ge 3$ have property (T).\n",
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