Strong approximation in random towers of graphs

TL;DR

This paper explores probabilistic analogues of strong approximation phenomena in arithmetic groups, demonstrating that random towers of graphs exhibit expansion and large subgroup closures with high probability.

Contribution

It introduces a probabilistic framework for strong approximation in graph towers, showing that random constructions yield expanders and large subgroup closures almost surely.

Findings

Random towers of graphs are almost surely expanders.

Subgroups have positive Hausdorff dimension in the automorphism group.

Connected components of the graphs are bounded in number.

Abstract

The term "strong approximation" is used to describe phenomena where an arithmetic group as well as all of its Zariski dense subgroups have a large image in the congruence quotients. We exhibit analogues of such phenomena in a probabilistic, rather than arithmetic, setting. Let T be the binary rooted tree, Aut(T) its automorphism group. To a given m-tuple a = {a_1,a_2,...,a_m} in Aut(T), we associate a tower of 2m-regular Schreier graphs ...X_n-->X_{n-1}-->...-->X_0. The vertices of X_n are the n^{th} level of the tree and two such are connected by an edge if a generator takes one to the other. When {a_i} are independent Haar-random elements of Aut(T) we retrieve the standard model for iterated random 2-lifts studied, for example by Bilu-Linial. If w={w_1,w_2,...,w_l} are words in the free group F_m, the random substitutions w(a) := {w_1(a),...,w_l(a)} give rise to new models for random towers of 2l-regular graphs: ...Y_n-->Y_{n-1}-->...-->Y_0. With the above notation, the following hold almost surely, for every non cyclic subgroup D in F_m: (i) the graphs $Y_n$ have a bounded number of connected components, (ii) these connected components form a family of expander graphs, (iii) the closure of D has positive Hausdorff dimension as a subgroup of the (metric) group Aut(T).