Kesten's theorem for Invariant Random Subgroups

TL;DR

This paper extends Kesten's theorem to invariant random subgroups, showing spectral radius inequalities for nonamenable subgroups and implications for Ramanujan graphs and Benjamini-Schramm convergence.

Contribution

It generalizes Kesten's theorem from normal to invariant random subgroups and explores spectral properties and graph convergence implications.

Findings

Spectral radius of random walks on nonamenable invariant random subgroups is strictly less.

Sequences of finite quotients approximating certain linear groups converge in Benjamini-Schramm sense.

Infinite Ramanujan Schreier graphs have large girth.

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.