Haagerup property

In

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 ${\displaystyle G}$ be a

1. There is a
   negative definite function
   ${\displaystyle \Psi \colon G\to \mathbb {R} ^{+}}$.
2. ${\displaystyle G}$ has the Haagerup approximation property, also known as Property ${\displaystyle C_{0}}$: there is a sequence of normalized continuous positive-definite functions ${\displaystyle \phi _{n}}$ which vanish at infinity on ${\displaystyle G}$ and converge to 1
   compact subsets
   of ${\displaystyle G}$.
3. There is a
   strongly continuous unitary representation
   of ${\displaystyle G}$ which weakly contains the trivial representation and whose matrix coefficients vanish at infinity on ${\displaystyle G}$.
4. There is a proper continuous affine isometric action of ${\displaystyle G}$ 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
  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 ${\displaystyle \mathbb {R} }$-trees
• Coxeter groups
• Amenable groups
• Groups acting properly on
  CAT(0) cubical complexes

• Cherix, Pierre-Alain; Cowling, Michael; Jolissaint, Paul; Julg, Pierre; Valette, Alain (2001), Groups with the Haagerup property. Gromov's a-T-menability., Progress in Mathematics, vol. 197, Basel: Birkhäuser Verlag,
  MR 1852148