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  "body": "\n## Definition\n\nLet $G$ be a locally compact group, and let $\\pi$ be a unitary representation of $G$ on a [[Hilbert space|HM2HHE3J51]] $\\mathcal{H}$.\nFor a [[compact|0TVJZYB6SH]] set $Q \\subseteq G$ and $\\varepsilon \u003e 0$, a unit vector $\\xi \\in \\mathcal{H}$ is *$(Q, \\varepsilon)$-invariant* if $\\norm{\\pi(g) \\xi - \\xi} \u003c \\varepsilon$ for all $g \\in Q$.\nA vector $\\xi$ is *invariant* if $\\pi(g) \\xi = \\xi$ for all $g \\in G$.\nThe group $G$ has *property (T)* if there exist a compact set $Q \\subseteq G$ and $\\varepsilon \u003e 0$ with the following property.\nEvery unitary representation of $G$ with a $(Q, \\varepsilon)$-invariant unit vector has a nonzero invariant vector.\n\n## Remarks\n\nExample: every compact group $G$ has property (T), with $Q = G$ and $\\varepsilon = \\sqrt{2}$.\nIf $\\norm{\\pi(g)\\xi - \\xi} \u003c \\sqrt{2}$ for all $g$, then $\\mathrm{Re} \\ip{\\pi(g)\\xi}{\\xi} \u003e 0$ for all $g$.\nThe average $\\eta = \\int_G \\pi(g) \\xi \\, dg$ over the normalised Haar measure is invariant, and $\\mathrm{Re} \\ip{\\eta}{\\xi} \u003e 0$, so $\\eta \\neq 0$.\n\nNon-example: $\\Z$ does not have property (T).\nHere compact sets are finite.\nIn the regular representation on $\\ell^2(\\Z)$, let $\\xi_N = N^{-1/2} \\mathbf{1}_{\\set{0, \\ldots, N-1}}$.\nThen $\\norm{\\pi(k)\\xi_N - \\xi_N} \\le \\sqrt{2\\abs{k}/N}$, which is small on any finite $Q$ when $N$ is large.\nBut a nonzero vector of $\\ell^2(\\Z)$ invariant under all shifts would have constant coordinates and infinite norm.\n\nEquivalently, the trivial representation is an isolated point of the unitary dual of $G$ in the Fell topology.\nFor a discrete group, $Q$ can be taken finite, and a discrete group with property (T) is finitely generated.\nThe groups $\\mathrm{SL}_n(\\R)$ and $\\mathrm{SL}_n(\\Z)$ have property (T) for $n \\ge 3$, but not for $n = 2$.\n",
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