Haagerup property

In mathematics, the Haagerup property, also known as Gromov's a-T-menability, is a property of groups that is a strong negation of Kazhdan's property (T). Property (T) is considered a representation-theoretic form of rigidity, so the Haagerup property may be considered a form of strong nonrigidity; see below for details.

The Haagerup property is interesting to many fields of mathematics, including harmonic analysis, representation theory, operator K-theory, and geometric group theory.

Perhaps its most impressive consequence is that groups with the Haagerup Property satisfy the Baum-Connes conjecture and the related Novikov conjecture. Groups with the Haagerup property are also uniformly embeddable into a Hilbert space.

Definitions

Let $G$ be a second countable locally compact group. The following properties are all equivalent, and any of them may be taken to be definitions of the Haagerup property:

#There is a proper continuous conditionally negative definite function $Psicolon\; G\; o\; Bbb\{R\}^+$.
#$G$ has the Haagerup approximation property, also known as Property $C\_0$: there is a sequence of normalized continuous positive definite functions $phi\_n$ which vanish at infinity on $G$ and converge to 1 uniformly on compact subsets of $G$.
#There is a strongly continuous unitary representation of $G$ which weakly contains the trivial representation and whose matrix coefficients vanish at infinity on $G$.
#There is a proper continuous affine action of $G$ on a Hilbert space.
#For every closed subgroup $H$ of $G$ such that the pair $(G,H)$ has relative property (T), $H$ is compact.

Examples

There are many examples of groups with the Haagerup property, most of which are geometric in origin. The list includes:

*All compact groups (trivially). Note all compact groups also have property (T). The converse holds as well: if a group has both property (T) and the Haagerup property, then it is compact.
*SO(n,1)
*SU(n,1)
*Groups acting properly on trees or on $Bbb\{R\}$-trees
*Coxeter groups
*Amenable groups
*Groups acting properly on CAT(0) cubical complexes

References

*Cherix, Cowling, Jolissaint, Julg, and Valette (2001). "Groups with the Haagerup Property (Gromov's a-T-menability)"