Kesten's theorem for invariant random subgroups

Abstract

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. © 2014.