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 = {a1, a2, . . . , am} ∈ Aut(T ) we associate a tower of 2m-regular Schreier graphs . . . → Xn → Xn−1 → . . . → X0. The vertices of Xn are the n th level of the tree and two such x, y ∈ Xn are connected by an edge if y = x ai or if x = yi for some i. When {ai} ⊂ Aut(T ) are independent Haar-random elements we retrieve the standard model for iterated random 2-lifts studied in [BL04],[AL02],[ALM02]. If w = {w1, w2, . . . , wl} ⊂ Fm are words in the free group, the random substitutions w(a) := {w1(a), . . . , wl(a)} give rise to new models for random towers of 2l-regular graphs. . . . → Yn → Yn−1 → . . . → Y0. Theorem A. With the above notation, the following hold almost surely, whenever ∆ := 〈w〉 is a non-cyclic subgroup of Fm: • The graphs Yn have a bounded number of connected components, • these connected components form a family of expander graphs. • the closure ∆ has positive Hausdorff dimension as a subgroup of the (metric) group Aut(T ). Some of this is generalized to more general trees.