Approximation properties for discrete groups.

Approximation properties for groups lie at the intersection of functional analysis and geometric group theory, which are central areas in modern mathematics. Such properties often act as bridges between these two fields. Probably, the most wellknown approximation property is amenability, which was introduced in 1929 by von Neumann in his work on the Banach-Tarski paradox and has been a key ingredient in several groundbreaking results since. This research proposal is related to a much weaker approximation property, namely the Approximation Property of Haagerup and Kraus (AP). Only in 2010, the first examples of groups without the AP were found, namely (lattices in) SL(n,R) for n > 2. My work with Haagerup gave rise to a much larger class of examples of groups without this property. The objectives are focused around the AP for discrete groups. The first one is to prove new permanence properties for the AP. Secondly, in an ongoing work, Haagerup and I formulate a natural obstruction to the AP. I want to formulate an analogue of this obstruction in the setting of von Neumann algebras, which form an important class of algebras of bounded operators on a Hilbert space. The third objective is to investigate whether the methods recently developed by Lafforgue and de la Salle to study noncommutative Lp-spaces can provide information about the open problem whether all p-convolvers are pseudomeasures. The essential novelty of this approach is that it is inherently p-dependent.