Strong approximation in random towers of graphs

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).