Actions of higher rank groups on uniformly convex Banach spaces

2023/03/02 by Tim de Laat, Mikael de la Salle, de Laat, Tim +1 · 3 citations
Mathematics · #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2303.01405

openalex publication_date 2023/03/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

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.

Cited by

• Higher property T and below-rank phenomena of lattices
• Discrete Poincaré inequalities and universal approximators for random graphs
• Geometric Banach property (T) for metric spaces via Banach representations of Roe algebras

Related

• On fixed points and uniformly convex spaces
• Banach fixed point property for Steinberg groups over commutative rings
• Superrigidity, generalized harmonic maps and uniformly convex spaces
• Strong Banach property (T), after Vincent Lafforgue
• Hyperbolic rigidity of higher rank lattices