Haagerup property

Let ${\displaystyle G}$ 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 ${\displaystyle \Psi \colon G\to \mathbb {R} ^{+}}$.
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 uniformly on 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.

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 ${\displaystyle \mathbb {R} }$-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, doi:10.1007/978-3-0348-8237-8, ISBN 3-7643-6598-6, MR 1852148