On Haagerup and Kazhdan properties

Abstract

In the first chapter, we characterize p-adic linear algebraic groups with the Haagerup Property. We also characterize connected Lie groups having the Haagerup Property viewed as discrete groups, and we provide an example of a finitely presented group not having the Haagerup Property, but having no infinite subgroup with relative Property (T). This example motivates the introduction in the second chapter of the relative Property (T) for an arbitrary subset of a locally compact group. In a connected Lie group, we characterize the subsets with relative Property (T). We introduce a notion of resolutions so as to extend the latter results to lattices in connected Lie groups. In the third chapter, we characterize connected Lie groups having a dense finitely generated subgroup with Property (T). In the fourth chapter, we provide an example of a finitely presented group having Property (T), non-Hopfian, and with infinite outer automorphism group. In the fifth chapter, we characterize locally nilpotent groups for which all unitary representations have vanishing reduced 1-cohomology. In the sixth chapter, we show that if F is a finite perfect group, and X is any set, then the unrestricted direct product FX is strongly bounded. This means that it has no isometric action on any metric space with unbounded orbits. Finally in the seventh chapter, we collect several short notes, including a list of open questions.