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arXiv · 2303.01405

Actions of higher rank groups on uniformly convex Banach spaces

Tim de Laat·Mikael de la Salle
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Abstract

We prove that all isometric actions of higher rank simple Lie groups and their lattices on arbitrary uniformly convex Banach spaces have a fixed point. This vastly generalises a recent breakthrough of Oppenheim. Combined with earlier work of Lafforgue and of Liao on strong Banach property (T) for non-Archimedean higher rank simple groups, this confirms a long-standing conjecture of Bader, Furman, Gelander and Monod. As a consequence, we deduce that sequences of Cayley graphs of finite quotients of a higher rank lattice are super-expanders.

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Tim de Laat, Mikael de la Salle. 2023-03-02. Actions of higher rank groups on uniformly convex Banach spaces. https://arxiv.org/abs/2303.01405

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