15 February 2014 Kesten’s theorem for invariant random subgroups
Miklós Abért, Yair Glasner, Bálint Virág
Duke Math. J. 163(3): 465-488 (15 February 2014). DOI: 10.1215/00127094-2410064


An invariant random subgroup of the countable group Γ is a random subgroup of Γ whose distribution is invariant under conjugation by all elements of Γ. We prove that for a nonamenable invariant random subgroup H, the spectral radius of every finitely supported random walk on Γ is strictly less than the spectral radius of the corresponding random walk on Γ/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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Miklós Abért. Yair Glasner. Bálint Virág. "Kesten’s theorem for invariant random subgroups." Duke Math. J. 163 (3) 465 - 488, 15 February 2014. https://doi.org/10.1215/00127094-2410064


Published: 15 February 2014
First available in Project Euclid: 11 February 2014

zbMATH: 1344.20061
MathSciNet: MR3165420
Digital Object Identifier: 10.1215/00127094-2410064

Primary: 20F69 , 22D40
Secondary: 05C81 , 35P20 , 53C24

Rights: Copyright © 2014 Duke University Press


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Vol.163 • No. 3 • 15 February 2014
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