The central notion of this project is the Haagerup property which, in recent years, has found applications ranging from K-theory of C*-algebras, to rigidity theory, to geometric group theory. A group is said to have the Haagerup property if it admits a proper affine action on a Hilbert space, and a deep result of Higson and Kasparov asserts that a Haagerup group satisfies the Baum Connes conjecture. In this project, Lp generalizations of this concept will be studied quantitatively, together with cohomological aspects of the theory. It is for instance planned to attack a Gromov conjecture about the vanishing of reduced Lp-cohomology on amenable groups. More generally, it is proposed to determine for which amenable groups the first reduced cohomology with values in mixing (resp. weakly mixing) Lp-representation vanishes. In particular, this will produce new geometric invariants for a large class of groups.
Looking for quantities that remain invariant under certain transformations is one the main objectives in mathematics as in many scientific fields like for example, physics, information theory or finance. The principle behind geometric group theory is that many properties of a space, such as its topology or its geometry, can be revealed by studying its set of symmetries. This set has a mathematical structure called a group structure. The wealth of this approach results from the interactions between the algebraic properties of this structure and its geometric properties. In this project, it is proposed to study new geometric invariants of groups, and to relate them with their algebraic structure. Groups of matrices, that appear as fundamental objects in physics, will have a central position in this study.
