Duke Mathematical Journal

Kesten’s theorem for invariant random subgroups

Miklós Abért, Yair Glasner, and Bálint Virág

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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.

Article information

Duke Math. J. Volume 163, Number 3 (2014), 465-488.

First available in Project Euclid: 11 February 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F69: Asymptotic properties of groups 22D40: Ergodic theory on groups [See also 28Dxx]
Secondary: 05C81: Random walks on graphs 35P20: Asymptotic distribution of eigenvalues and eigenfunctions 53C24: Rigidity results


Abért, Miklós; Glasner, Yair; Virág, Bálint. Kesten’s theorem for invariant random subgroups. Duke Math. J. 163 (2014), no. 3, 465--488. doi:10.1215/00127094-2410064. https://projecteuclid.org/euclid.dmj/1392128875.

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