Kesten's theorem for Invariant Random Subgroups

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.Comment: 19 page

L^p Norms of Eigenfunctions on Regular Graphs and on the Sphere

Abstract We prove upper bounds on the $L^p$ norms of eigenfunctions of the discrete Laplacian on regular graphs. We then apply these ideas to study the $L^p$ norms of joint eigenfunctions of the Laplacian and an averaging operator over a finite collection of algebraic rotations of the two-sphere. Under mild conditions, such joint eigenfunctions are shown to satisfy for large $p$ the same bounds as those known for Laplace eigenfunctions on a surface of non-positive curvature.</jats:p

The measurable Kesten theorem

We give explicit estimates between the spectral radius and the densities of short cycles for finite d-regular graphs. This allows us to show that the essential girth of a finite d-regular Ramanujan graph G is at least c log log |G|. We prove that infinite d-regular Ramanujan unimodular random graphs are trees. Using Benjamini-Schramm convergence this leads to a rigidity result saying that if most eigenvalues of a d-regular finite graph G fall in the Alon-Boppana region, then the eigenvalue distribution of G is close to the spectral measure of the d-regular tree. Kesten showed that if a Cayley graph has the same spectral radius as its universal cover, then it must be a tree. We generalize this to unimodular random graphs.Comment: The previous, longer version 1 has been split in two parts: the present paper, and a more group-theoretic one with the title "Kesten's theorem for Invariant Random Subgroups

Actions and Invariants of Residually Finite Groups: Asymptotic Methods

The workshop brought together experts in finite group theory, L2-cohomology, measured group theory, the theory of lattices in Lie groups, probability and topology. The common object of interest was residually finite groups, that each field investigates from a different angle

Ramanujan Graphs With Small Girth

We construct an infinite family of (q +1) regular Ramanujan graphs X n of girth 1. We also give covering maps X n+1 ! X n such that the minimal common covering of all the X n &apos;s is the universal covering tree