Groups with the Haagerup Property
A second countable locally compact group has the Haagerup property if it has a proper continuous isometric action on an affine Hilbert space. The class of groups with this property is quite broad: examples include compact groups, SO(n, 1), SU(n, 1), Coxeter groups, free groups and amenable groups.
The book contains a series of papers concerning these groups, starting with a short introduction by A. Valette, and various equivalent definitions by P. Jolissaint. A geometric proof of the Haagerup property for SO(n, 1) and SU(n, 1) is given by P. Julg, and the classification of groups with the Haagerup property is presented by P.-A. Cherix, M. Cowling and A. Valette. The classification asserts that such a group is locally isomorphic to a direct product of an amenable group with a finite number of copies of SO(n, 1) and SU(n, 1) (dimensions can vary). M. Cowling’s studies groups with the radial Haagerup property. Some hereditary results (mainly for discrete groups) are discussed in the paper by P. Jolissaint, P. Julg and A. Valette. The book ends with a list of open problems prepared by A. Valette.