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SUMMARY:HIher-rank lattices and uniformly convex Banach spaces - Mikael de
  la Salle (Université Claude Bernard Lyon 1)
DTSTART:20250707T104500Z
DTEND:20250707T114500Z
UID:TALK232807@talks.cam.ac.uk
DESCRIPTION: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 l
 attices. In a seminal work in the study of group actions on Banach spaces\
 , Bader\, Furman\, Gelander\, and Monod conjectured that every action by i
 sometries of $\\Gamma$ on a uniformly convex Banach space has a fixed poin
 t. This conjecture was proven by Lafforgue and Liao for polynomials. The c
 ase of integers took longer to resolve (joint work with Tim de Laat\, foll
 owing a breakthrough by Izhar Oppenheim). I will present the similarities 
 and the differences between these two proofs. For those in the audience wh
 o 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 i
 n nilpotent subgroups.
LOCATION:Seminar Room 1\, Newton Institute
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