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.

The measurable Kesten theorem

“…As the readership of this result is expected to be different from that of the current paper (and the current paper is already long), we decided to publish it in a separate article, see [1].…”

“…We give the following two quantitative versions of Theorem 5. For infinite dregular unimodular random graphs (1) E log ρ(G) − log ρ(…”

Universal groups of intermediate growth and their invariant random subgroups

“…Observe that the Borel σ-algebra on S(G) is generated by sets of the form It is known (see [AGV12]) that every IRS of a finitely generated group arises from a measure preserving action on a Borel probability space (X, µ). (1) the group G acts transitively on the levels of T d , (2) the action of G on ∂T d is minimal (i.e., orbits are dense), (3) the action of G on ∂T d is ergodic with respect to the uniform Bernoulli measure on ∂T d .…”

“…A more general problem is the identification of the simplex of invariant probability measures of the system (Φ, S(G)) where Φ is a subgroup of the group Aut(G) of automorphisms of G (see [AGV12,Bow12,Ver12]). A closely related problem is the study of invariant measures on the space of rooted Schreier graphs of G, with G acting by change of the root.…”

Invariant random subgroups of linear groups

“…The term "IRS" was introduced in a pair of joint papers with Abért and Virág [AGV13,AGV14], however the paper of Stuck-Zimmer [SZ94] is quite commonly considered as the first paper on this subject. That paper provides a complete classification of IRS in a higher rank simple Lie group G, by showing that every ergodic IRS is supported on a single orbit (i.e.…”