31.12.07 - On Haagerup and Kazhdan properties. Thèse EPFL, n° 3438 (2006). Dir.: Prof. P. Buser

"Pour une thèse extraordinaire, accomplie en un temps record, qui résout des problèmes restés longtemps ouverts de la théorie des représentations des groupes algébriques"

On Haagerup and Kazhdan properties.

The study of isometric actions of groups has proved to be a fundamental tool in representation theory as in geometric group theory. A locally compact group has the Haagerup Property if it possesses an isometric action on a Hilbert space, which is metrically proper. At the opposite, it has the Kazhdan Property if every such action has a fixed point. This work is, in a large part, devoted to interpolate between these two properties, by introducing the relative Kazhdan Property for any pair (G,X), X being any subset of the locally compact group G.

This generalizes Margulis' relative Kazhdan Property for pairs (G,H) where H is a subgroup of G. However, we give the first known example of a group G having unbounded subsets with the relative Kazhdan Property (and therefore failing to have the Haagerup Property), but none of whose can be chosen as a subgroup. A thorough study of this relative Property is carried out for Lie groups and their lattices.