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Kesten’s theorem for invariant random subgroups. (English) Zbl 1344.20061

Summary: 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.

MSC:

20F69 Asymptotic properties of groups
22D40 Ergodic theory on groups
05C81 Random walks on graphs
60G50 Sums of independent random variables; random walks
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