Informal Group Action Seminar

In his study of the Baum-Connes conjecture, Vincent Lafforgue was lead to study Banach space representations of SL(n>2,F) (for F a local field) which are not by isometries but by operators of slowly growing norm. Classical tools to study unitary representations of higher rank groups are not efficient in this context, so he had to develop a novel approach, which allowed him to prove that cocompact lattices in SL(n>2,F) have a form of property (T) adapted to this context, that he called propriété (T) renforcée Banachique. This approach to higher rank rigidity has already had several applications, notably to metric geometry (construction of super-expanders) or more recently in the study of actions of higher rank lattices on manifolds (Zimmer's programm). I am convinced that these ideas will be useful in other contexts. In my talk, I will present this approach, as well as some recent generalizations which allow in particular to generalize Lafforgue's results on strong property (T) to non cocompact lattices as well, for example to SL(n,Z). This involves the study of some representation-like objects of higher rank groups, in which one is only allowed to compose only once.