Property (T) and all that

In these talks I’ll try to explain these classes of groups so peculiarly separated by chance. I’ll start in § 1 with amenable groups, which have been studied the longest (since von Neumann in the 1920s, who was investigating the Banach–Tarski paradox as Henry explained last term), and are a very natural class of groups to look at. Then I’ll move on in § 2 to an enormous superset of the amenable groups— the Haagerup groups, or in Gromov’s vivid terminology, the a-T-menable groups. § 3 will explain what the T in a-T-menable is all about, and in § 4 we’ll see an application to the construction of families of expander graphs. One might ask why someone interested primarily in discrete groups should consider these properties that seem to have more to do with continuous representation theory. Aside from the celebrated rigidity results linking the two approaches to group theory, these groups have also fallen into the ambit of the Baum–Connes industry. As we shall shortly see in our K homology study group, Higson and Kasparov have proved the Baum–Connes conjecture with coefficients for a-T-menable groups, while current methods are badly adapted to deal with property (T) groups: in particular, the conjecture is unknown for SL(n,Z) with n ≥ 3. However, in these talks I’ll adopt a far more down-to-earth point of view: it turns out that property (T), and its ‘opposite’ property, a-T-menability, are closely related to actions on spaces with walls, which are of considerable interest to our group as they include R-trees and CAT(0) cube complexes. I’ll try to bring out these links. I’d like to thank Yves de Cornulier for suggesting corrections to a previous version of these notes, and for pointing out some interesting recent results in the area.