Abstract

Consider the group $\Gamma= \mathrm{SL}_3(A)$, where $A$ is one of two rings: the integers, or the polynomials over a finite field. These two groups are emblematic examples in a larger family: higher-rank lattices. In a seminal work in the study of group actions on Banach spaces, Bader, Furman, Gelander, and Monod conjectured that every action by isometries of $\Gamma$ on a uniformly convex Banach space has a fixed point. This conjecture was proven by Lafforgue and Liao for polynomials. The case of integers took longer to resolve (joint work with Tim de Laat, following a breakthrough by Izhar Oppenheim). I will present the similarities and the differences between these two proofs. For those in the audience who do not care about Banach spaces, this will be a new proof of Kazhdan’s theorem that $\Gamma$ has property (T), where all the analysis is done in nilpotent subgroups.