Yair Glasner

Kesten's theorem for Invariant Random Subgroups

An invariant random subgroup of the countable group is a ran dom subgroup of whose distribution is invariant under conjuga tion by all elements of . We prove that for a nonamenable invariant random subgroup H, the spectral radius of every finitely supported random walk on i s strictly less than the spectral radius of the corresponding random walk on /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

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.

Amenable actions, free products and a fixed point property

We investigate the class of groups admitting an action on a set with an invariant mean. It turns out that many free products admit such an action. We give a complete characterisation of such free products in terms of a strong fixed point property.

A two-dimensional version of the Goldschmidt–Sims conjecture☆

Abstract The Goldschmidt–Sims conjecture asserts that there is a finite number of (conjugacy classes of) edge transitive lattices in the automorphism group of a regular tree with prime valence. We prove a similar theorem for irreducible lattices, transitive on the 2-cells of the product of two regular trees of prime valences.

Generic groups acting on regular trees

Let T be a k-regular tree (k ≥ 3) and A = Aut(T) its automorphism group. We analyze a generic finitely generated subgroup Γ of A. We show that Γ is free and establish a trichotomy on the closure Γ of Γ in A. It turns out that Γ is either discrete, compact or has index at most 2 in A.

Invariant random subgroups of linear groups

AbstractLet Γ < GLn(F) be a countable non-amenable linear group with a simple, center free Zariski closure. Let Sub(Γ) denote the space of all subgroups of Γ with the compact, metric, Chabauty topology. An invariant random subgroup (IRS) of Γ is a conjugation invariant Borel probability measure on Sub(Γ). An IRS is called non-trivial if it does not have an atom in the trivial group, i.e. if it is non-trivial almost surely. We denote by IRS0(Γ) the collection of all non-trivial IRS on Γ.Theorem 0.1: With the above notation, there exists a free subgroupF < Γ and a non-discrete group topology on Γ such that for everyμ ∈ IRS0(Γ) the following properties hold: μ-almost every subgroup of Γ is openF ·Δ = Γ for μ-almost every Δ ∈ Sub(Γ).F ∩ Δ is infinitely generated, for every open subgroup. In particular, this holds for μ-almost every Δ ∈ Sub(Γ).The map Φ: (Sub(Γ), μ) → (Sub(F),Φ*μ) Δ → Δ ∩ Fis an F-invariant isomorphism of probability spaces.A more technical version of this theorem is valid for general countable linear groups. We say that an action of Γ on a probability space, by measure preserving transformations, is almost surely non-free (ASNF) if almost all point stabilizers are non-trivial.Corollary 0.2: Let Γ be as in the Theorem above. Then the product of finitely many ASNF Γ-spaces, with the diagonal Γ action, is ASNF.Corollary 0.3: Let Γ < GLn(F) be a countable linear group, A Δ Γ the maximal normal amenable subgroup of Γ — its amenable radical. If μ ∈ IRS(Γ) is supported on amenable subgroups of Γ, then in fact it is supported on Sub(A). In particular, if A(Γ) = then Δ = , μ almost surely.

Sharply 2-Transitive Linear Groups

Abstract A group Γ is sharply 2-transitive if it admits a faithful permutation representation that is transitiveand free on pairs of distinct points. Conjecturally, for all such groups there exists a near-field N (i.e. askew field that is distributive only from the left, see Definition 2.10) such that Γ ∼= N × ⋉N. This iswell known in the finite case. We prove this conjecture when Γ < GL n (F) is a linear group, where F isany field with char(F) 6= 2 and that p -char(Γ) 6= 2 (see Definition 2.2). 1 Introduction A sharply k-transitive group is, by definition, a permutation group which acts transitively and freelyon ordered k-tuples of distinct points. Quite early on it was realized that k is very limited; in his 1872paper, [10], Jordan proved that finite sharply k-transitive groups with k ≥ 4 are either symmetric,alternating or one of the Mathieu group M 11 ,M 12 . In the infinite case it was proved by J. Tits, in [16],and M. Hall, in [8], that k ≤ 3 for every infinite sharply k-transitive group.Sharply 2-transitive groups attracted the attention of algebraists for many years because they lieon the borderline of permutation group theory and abstract algebraic structures. It is easy to see thatif K is a skew field then the semi-direct product K

RAMANUJAN GRAPHS WITH SMALL GIRTH.

We construct an infinite family of (q + 1)-regular Ramanujan graphs Xn of girth 1. We also give covering maps Xn+1 → Xn such that the minimal common covering of all the Xn’s is the universal covering tree.

Automorphism Groups of Trees Acting Locally with Affine Permutations

We consider the structure of groups that act on a pn-regular tree in a vertex transitive way with the local action (i.e. the action of the vertex stabilizer on the link) isomorphic to the group of affine transformations on a finite affine line.

Strong approximation in random towers of graphs

AbstractLet T=T2 be the rooted binary tree, Aut(T) =