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Local spectral gap in simple Lie groups and applications. (English) Zbl 1366.22004
One of the main motivations for this paper is to identify analogues of the results by J. Bourgain and A. Gamburd [ibid. 171, No. 1, 83–121 (2008; Zbl 1135.22010); J. Eur. Math. Soc. (JEMS) 14, No. 5, 1455–1511 (2012; Zbl 1254.43010)] and Y. Benoist and N. de Saxcé [Invent. Math. 205, No. 2, 337–361 (2016; Zbl 1357.22003)] that apply to general simple Lie groups. By analogy with the compact case, the authors introduce a notion of spectral gap for infinite measure preserving actions, which in the case of left translation actions on locally compact groups, implies a uniqueness property for its left Haar measures as finitely additive measures. Applications are given to the Banach-Ruziewicz problem, orbit equivalence rigidity, continuous and monotone expanders, and bounded random walks.

##### MSC:
 22D40 Ergodic theory on groups 22E30 Analysis on real and complex Lie groups 22E46 Semisimple Lie groups and their representations 28D05 Measure-preserving transformations 43A75 Harmonic analysis on specific compact groups
##### Keywords:
local spectral gap; measure preserving actions
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##### References:
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