Banach property (T) for SLn(Z) and its applications

• 成果类型: Article
• 署名作者: Oppenheim, Izhar
• 署名单位: Ben-Gurion University of the Negev
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-023-01211-7
• 发表日期: 2023
• 页码: 893-930
• 关键词: fixed-point properties; warped cones; RIGIDITY; lattices

摘要

We prove that a large family of higher rank simple Lie groups (including SLn(R) for n >= 3) and their lattices have Banach property (T) with respect to all super-reflexive Banach spaces. Two consequences of this result are: First, we deduce Banach fixed point properties with respect to all super-reflexive Banach spaces for a large family of higher rank simple Lie groups. For example, we show that for every n >= 4, the group SLn(R) and all its lattices have the Banach fixed point property with respect to all super-reflexive Banach spaces. Second, we settle a long standing open problem and show that the Margulis expanders (Cayley graphs of SLn(Z/mZ) for a fixed n >= 3 and m tending to infinity) are super-expanders. All of our results stem from proving Banach property (T) for SL3(Z). Our method of proof for SL3(Z) relies on a novel proof for relative Banach property (T) for the uni-triangular subgroup of SL3(Z). This proof of relative property (T) is new even in the classical Hilbert setting and is interesting in its own right.