Strong approximation in random towers of graphs

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3 Scopus citations


Let T=T 2 be the rooted binary tree, Aut(T) = limAutn(T) its automorphism group and Ψn: Aut(T)→Autn(T) the restriction maps to the first n levels of the tree. If L n is the the n th level of the tree then Autn(T) < Sym(L n) can be identified with the 2-Sylow subgroup of the symmetric group on 2n points. Consider a random subgroup Γ:= 〈a〉= 〈a 1, a 2,..., a m〉 ∈ Aut(T)m generated by m independent Haar-random tree automorphisms. Theorem A. The following hold, almost surely, for every non-cyclic subgroup Δ < Γ: • The closure Δ̄ < Aut(T) has positive Hausdorff dimension. In other word (Formula presented.)• The number of orbits of Δ on L n is bounded, independent of n. • If Δ=〈w〉=〈w 1, w 2,... w l〉 is finitely generated then the connected components of the Schreier graphs Yn = G(Δ, w, Ln) coming from the action of Δ on the different levels of the tree form a family of expander graphs.

Original languageEnglish
Pages (from-to)139-172
Number of pages34
Issue number2
StatePublished - 1 Jan 2014

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Computational Mathematics


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