Local spectral gap in simple Lie groups and applications

International audienceWe 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 of Γ on 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 [BG06, BG10], and Benoist and de Saxcé [BdS14]. 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