## 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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### References:

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