@MISC{Glasner09strongapproximation, author = {Yair Glasner}, title = {STRONG APPROXIMATION IN RANDOM TOWERS OF GRAPHS}, year = {2009} }

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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 = {a1, a2,..., am} ∈ Aut(T) m 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 = y ai 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.