A group which act properly on a CAT(0) cubical complex, while not necessarily amenable, satisfies several weak forms of amenability: such a group is a-T-menable, weakly amenable and K-theoretically amenable. In the talk, based on joint work with J. Brodzki and N. Higson, I will describe a proof of K-amenability which finds its roots in earlier work of P. Julg and A. Valette on groups acting on trees. I will focus on the geometric constructions involved, and will keep the analytic complications to a minimum

Important classes of locally compact groups are characterized by their actions by affine isometries on Hilbert spaces (groups with Kazhdan's property, a-T-menable groups aka groups with the Haagerup property). We will be interested on the question of irreducibility of such actions, in the sense that the only non empty closed invariant affine subspace is the whole space. This notion was extensively studied in a recent joint work of T. Pillon, A. Valette and myself. We will report on this work as well as on some further results. Special attention will be paid to affine actions whose linear part is a factorial representation, that is, a representation which generates a factor in the von Neumann algebra sense

We will present some results about Kazhdan sets in topological groups using functional and harmonic analysis methods. We will discuss in particular a question, raised by Shalom, about Kazhdan sets and equidistribution properties. This is joint work with Sophie Grivaux

The talk will focus on the following results: a finitely generated group quasi-isometric to the fundamental group of a compact 3–manifold or to a finitely generated Kleinian group contains a finite index subgroup isomorphic to the fundamental group of a compact 3–manifold or to a finitely generated Kleinian group

Every fixed-point-free isometric action of a semisimple Lie group G is proper.

I will explain this in my talk. Later I will elaborate on further generalizations of this fact.

For example, the image of G under any continuous homomorphism into a topological group is closed. Another application is given by describing explicitly the WAP compactification of G (a notion that will be defined in the talk). The latter will be related to the decay property of matrix coefficients of reflexive representations.

The talk will be based on a joint work with Tsachik Gelander

The following natural question arises from Shalom's innovational work (1999, Publ.IHES) on Kazhdan's property (T). ``Can we establish an `intrinsic' criterion to synthesize relative fixed point properties into the whole fixed point property without assuming `Bounded Generation'?'' This talk is aimed to present a resolution to this question in the affirmative. Our criterion works for ones with respect to certain classes of Busemann Non-Positively Curved spaces. It, moreover, suggests a further step toward constructing super-expanders from finite simple groups of Lie type.

Higman’s embedding theorem says that any countable group can be embedded in a group generated by two elements. The relative version of this asks: given a countable subgroup H of a large group G, does H always lie in a finitely generated subgroup of G? (Of course, the answer should depend on G). This talk will answer this question for some interesting classes of groups, and discuss the related notions of strong boundedness (the property that every action of G by isometries on any metric space has all orbits bounded) and strong distortion. Far from pathological examples, the groups we consider are all groups of homeomorphisms or diffeormophisms of manifolds; where boundedness and distortion of subgroups of homeomorphisms can say something about the dynamics of their actions on the manifold. This is new joint work with F. Le Roux