Tim de Laat

Actions of higher rank groups on uniformly convex Banach spaces

Fixed point properties for isometric group actions on Banach spaces are fundamental rigidity properties which can be viewed as Banach space versions of Kazhdan's property (T). After an introduction to this topic, I will explain that all isometric actions of higher rank semisimple Lie/algebraic groups (over arbitrary local fields) and their lattices on arbitrary uniformly convex Banach spaces have a fixed point. This result, which is a combination of the Archimedean and the non-Archimedean case, confirmed a long-standing conjecture of Bader, Furman, Gelander and Monod. This talk is based on joint work with Mikael de la Salle.