---
tag: 5FG1BKZXKS
kind: concept
name: Haagerup property
msc: [22D55, 20F65]
sources:
  - url: https://en.wikipedia.org/wiki/Haagerup_property
same_as:
  wikidata: Q5636478
  wikipedia: Haagerup property
---

## Summary

A locally compact group has the Haagerup property when it has a unitary representation with almost invariant vectors whose matrix coefficients vanish at infinity.
Equivalently, it acts on a Hilbert space by affine isometries in a proper way; for this reason Gromov called such groups a-T-menable.
The property is a weak form of [[amenability|CEAT9BESKK]] and is opposite to [[property (T)|GTFZ277047]]: a group with both properties is compact.
Haagerup showed that free groups have it, and it is used in the Baum–Connes conjecture and in the theory of von Neumann algebras.

## Explanations

### L1 human

Imagine 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.
Groups with such an action have the Haagerup property.
For the integers, translating along a line already works: the element $n$ moves the origin to distance $\abs{n}$.
Free groups also have such actions, built from the tree on which they act.
Groups with property (T), such as $\mathrm{SL}_3(\Z)$, are rigid in the opposite way: every such action of them has a fixed point.