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.

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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