Notes of Alain Valette’s Lille lecture

Graphs of groups and the Haagerup property

joint with Yves Cornulier

1. The Haagerup property

1.1. Basics

Definition 1 ${G}$ has Haagerup property if it has a proper affine sometric action on some Hilbert space.

Gromov calls them a-T-menable.

Example 1 (of a-T-menable groups)

– Amenable groups.

– Automorphism groups of locally finite trees.

– Coxeter groups.

– Closed subgroups of ${SO(n,1)}$ and ${SU(n,1)}$.

– Countable subgroups of ${Gl(2,F)}$, ${F}$ a field (Guentner).

Example 2 (of non a-T-menable groups)

– Groups having Kazhdan’s property (T).

1.2. Why do we care ?

These form the largest class of group for which Kaplansky’s conjecture is known. Indeed, Haagerup ${\Rightarrow}$ strong Baum-Connes ${\Rightarrow}$ (if discrete and torsion free) Kaplansky’s conjecture.

2. Permanence properties

– Stable under taking closed subgroups.

– Stable under finite direct products.

Caveat: not stable under semi-direct products.

Example 3 ${Sl(2,{\mathbb Z}) \times {\mathbb Z}^2}$ has relative property (T), therefore every affine isometric action on a Hilbert space has ${{\mathbb Z}^2}$-invariant vectors.

Can replace ${Sl(2,{\mathbb Z})}$ with free group ${F_2}$. Note that the semi-direct product acts on a tree with stabilizers ${{\mathbb Z}^2}$.

Question. Let ${G}$ be a group acting on a tree with stabilizers which have Haagerup property. Find assumptions ensuring that ${G}$ have Haagerup property.

Here is a sample result.

Proposition 2 (Jolissaint-Julg-Valette 2001) Ok if edge stabilizers are finite.

Precise question: What if edge stabilizers are virtually cyclic ?

3. Graphs of groups

Definition 3

We fix ${n}$ and consider graphs of groups where all vertex and edge stabilizers are ${Z^n}$. Let ${G}$ be the corresponding fundamental group.

Such a group comes with an isometric action on a tree ${T}$ (the Bass-Serre tree), and an affine action on ${\mathbb{R}^n}$. Indeed, let ${N}$ be the normal subgroup generated by vertex groups. There is a natural map of ${N}$ to ${{\mathbb Z}^n}$, and so it acts by translations on ${\mathbb{R}^n}$. Let ${d}$ be the connectivity of the graph. Then ${G/N}$ is a free group of rank ${d}$. Pick representatives (edges unused in a maximal tree), to each of them, a virtual isomorphism of ${{\mathbb Z}^n}$ is associated.

Example 4 – For ${Sl(2,{\mathbb Z}) \times {\mathbb Z}^2}$, it is the obvious affine action.

– For Baumslag-Solitar groups ${BS(m,n)=< a,b | ab^m a^{-1} = b^n >}$, map ${b}$ to ${(x \rightarrow x+1)}$ and ${a}$ to ${(x \rightarrow m/n x)}$.

Theorem 4 The following are equivalent.

1. ${G}$ has the Haagerup property.

2. The closure ${C}$ of the image of ${G/N}$ in ${Gl(n,{\mathbb R})}$ is amenable.

Due to Gal and Januskiewicz 2003 for Baumslag-Solitar groups (in that case, ${d=1}$ and 2 is always true). They view these groups as lattices in products. This works in our generality: the action of ${G}$ on ${T \times \mathbb{R}^n}$ is proper. It turns ${G}$ into a cocompact lattice in the closed subgroup ${L=M \times ({\mathbb R}^n \times C)}$ of ${Aut(T) \times Aff({\mathbb R}^n)}$.

There remains to decide when ${\mathbb{R}^n \times C}$ is Haagerup.

Proposition 5 Let ${\Gamma}$ have Haagerup property. The following are equivalent.

1. ${\mathbb{R}^n \times \Gamma}$ has Haagerup property.

2. The closure of ${\Gamma}$ in ${Gl(n,{\mathbb R})}$ is amenable.

Uses Cornulier’s PhD (various characterizations of relative property (T)) and Furstenberg’s criterion for the existence of invariant measures.