HIher-rank lattices and uniformly convex Banach spaces

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• Mikael de la Salle (Université Claude Bernard Lyon 1)
• Monday 07 July 2025, 11:45-12:45
• Seminar Room 1, Newton Institute.

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NPCW06 - Non-positive curvature and applications

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.

This talk is part of the Isaac Newton Institute Seminar Series series.

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