Shear and moment diagram

In mathematics, the Haagerup property, named after Uffe Haagerup and 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:

1. There is a proper continuous conditionally negative definite function $\mathrm{\Psi }:G\to {\mathbb{ℝ}}^{+}$.
2. $G$ has the Haagerup approximation property, also known as Property $C_0$: there is a sequence of normalized continuous positive-definite functions ${\varphi }_n$ which vanish at infinity on $G$ and converge to 1 uniformly on compact subsets of $G$.
3. There is a strongly continuous unitary representation of $G$ which weakly contains the trivial representation and whose matrix coefficients vanish at infinity on $G$.
4. There is a proper continuous affine isometric action of $G$ on a Hilbert space.

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 $\mathbb{ℝ}$-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)