Local spectral gap in simple Lie groups and applications

Abstract

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 $$\Gamma \curvearrowright G$$ , whenever $$\Gamma $$ 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 Saxce (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 $$\Gamma $$ -invariant finitely additive measure defined on all bounded measurable subsets of G.