Graphs of groups and the Haagerup property
joint with Yves Cornulier
1. The Haagerup property
Definition 1 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 and .
– Countable subgroups of , 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 strong Baum-Connes (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 has relative property (T), therefore every affine isometric action on a Hilbert space has -invariant vectors.
Can replace with free group . Note that the semi-direct product acts on a tree with stabilizers .
Question. Let be a group acting on a tree with stabilizers which have Haagerup property. Find assumptions ensuring that 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
We fix and consider graphs of groups where all vertex and edge stabilizers are . Let be the corresponding fundamental group.
Such a group comes with an isometric action on a tree (the Bass-Serre tree), and an affine action on . Indeed, let be the normal subgroup generated by vertex groups. There is a natural map of to , and so it acts by translations on . Let be the connectivity of the graph. Then is a free group of rank . Pick representatives (edges unused in a maximal tree), to each of them, a virtual isomorphism of is associated.
Example 4 – For , it is the obvious affine action.
– For Baumslag-Solitar groups , map to and to .
Theorem 4 The following are equivalent.
1. has the Haagerup property.
2. The closure of the image of in is amenable.
Due to Gal and Januskiewicz 2003 for Baumslag-Solitar groups (in that case, and 2 is always true). They view these groups as lattices in products. This works in our generality: the action of on is proper. It turns into a cocompact lattice in the closed subgroup of .
There remains to decide when is Haagerup.
Proposition 5 Let have Haagerup property. The following are equivalent.
1. has Haagerup property.
2. The closure of in is amenable.
Uses Cornulier’s PhD (various characterizations of relative property (T)) and Furstenberg’s criterion for the existence of invariant measures.