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1 IMB - Institut de Mathématiques de Bordeaux 2 Mathematics Department; University of California San Diego

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 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.





Author: Rémi Boutonnet - Adrian Ioana - Alireza Salehi-Golsefidi -

Source: https://hal.archives-ouvertes.fr/



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