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  "body": "\n## Summary\n\nA locally compact group has the Haagerup property when it has a unitary representation with almost invariant vectors whose matrix coefficients vanish at infinity.\nEquivalently, it acts on a Hilbert space by affine isometries in a proper way; for this reason Gromov called such groups a-T-menable.\nThe property is a weak form of [[amenability|CEAT9BESKK]] and is opposite to [[property (T)|GTFZ277047]]: a group with both properties is compact.\nHaagerup showed that free groups have it, and it is used in the Baum–Connes conjecture and in the theory of von Neumann algebras.\n\n## Explanations\n\n### L1 human\n\nImagine a group acting on an infinite-dimensional space by rigid motions, so that points far out in the group move a given point far away.\nGroups with such an action have the Haagerup property.\nFor the integers, translating along a line already works: the element $n$ moves the origin to distance $\\abs{n}$.\nFree groups also have such actions, built from the tree on which they act.\nGroups with property (T), such as $\\mathrm{SL}_3(\\Z)$, are rigid in the opposite way: every such action of them has a fixed point.\n",
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