Kesten's theorem for Invariant Random Subgroups

Details Kesten's theorem for Invariant Random Subgroups Kesten's theorem for Invariant Random Subgroups

2012-01-17

arxiv.org/abs/1201.3399v1

Mathematics

Group Theory

19 pages

Scientific paper

An invariant random subgroup of the countable group {\Gamma} is a random subgroup of {\Gamma} whose distribution is invariant under conjugation by all elements of {\Gamma}. We prove that for a nonamenable invariant random subgroup H, the spectral radius of every finitely supported random walk on {\Gamma} is strictly less than the spectral radius of the corresponding random walk on {\Gamma}/H. This generalizes a result of Kesten who proved this for normal subgroups. As a byproduct, we show that for a Cayley graph G of a linear group with no amenable normal subgroups, any sequence of finite quotients of G that spectrally approximates G converges to G in Benjamini-Schramm convergence. In particular, this implies that infinite sequences of finite d-regular Ramanujan Schreier graphs have essentially large girth.

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