Local spectral gap in simple Lie groups and applications

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 Γ ↷ G , whenever Γ 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 Saxcé (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 Γ -invariant finitely additive measure defined on all bounded measurable subsets of G .