Kesten’s theorem for invariant random subgroups

Abstract

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.