Local spectral gap in simple Lie groups and applications

• 成果类型: Article
• 署名作者: Boutonnet, Remi; Ioana, Adrian; Golsefidy, Alireza Salehi
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Inria; Universite de Bordeaux; University of California System; University of California San Diego
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-016-0699-8
• 发表日期: 2017
• 页码: 715-802
• 关键词: distributing points; ii1 factors; invariant; SUBGROUPS; Operators; THEOREM; GROWTH

摘要

We introduce a novel notion of local spectral gap for general, possibly infinite, measure preserving actions. We establish local spectral gap for the left translation action , whenever is a dense subgroup generated by algebraic elements of an arbitrary connected simple Lie group G. This extends to the non-compact setting works of Bourgain and Gamburd (Invent Math 171:83-121, 2008; J Eur Math Soc (JEMS) 14:1455-1511, 2012), and Benoist and de Saxc, (Invent Math 205:337-361, 2016). We present several applications to the Banach-Ruziewicz problem, orbit equivalence rigidity, continuous and monotone expanders, and bounded random walks on G. In particular, we prove that, up to a multiplicative constant, the Haar measure is the unique -invariant finitely additive measure defined on all bounded measurable subsets of G.