Strong approximation in random towers of graphs

Abstract

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 } ∈ Aut(T ) m we associate a tower of 2m-regular Schreier graphs . . . → X n → X n−1 → . . . → X 0 .

The finitely generated Hausdorff spectra of a family of pro-p groups

“…These open normal subgroups yield a base of neighbourhoods of the identity and induce a translation-invariant metric on Γ which is given by d S (x, y) = inf |Γ : Γ i | −1 | x ≡ y (mod Γ i ) , for x, y ∈ Γ. This gives, for a subset U ⊆ Γ, the Hausdorff dimension hdim S Γ (U ) ∈ [0, 1] with respect to the filtration series S. Recently there has been much interest concerning Hausdorff dimensions in profinite groups, starting with the pioneering work of Abercrombie [1] and of Barnea and Shalev [3]; recent work includes for example [2,4,[7][8][9][10][12][13][14]16,18]. Barnea and Shalev [3] proved the following group-theoretic formula of the Hausdorff dimension with respect to S of a closed subgroup H of Γ as a logarithmic density:…”

Hausdorff dimensions in p-adic analytic groups

“…It is central to the subject of fractal geometry; compare [8]. More recently, the concept of Hausdorff dimension has led to fruitful applications in the context of profinite groups; e.g., see [4,3,6,2,12,7,10,11,9]. Let G be a countably based profinite group and fix a filtration series S of G, i.e., a descending chain G = G 0 ⊇ G 1 ⊇ .…”