Strong approximation in random towers of graphs

Abstract

Let T=T 2 be the rooted binary tree, Aut(T) = \(\mathop {\lim }\limits_ \leftarrow \)Aut n (T) its automorphism group and Ψ n : Aut(T)→Aut n (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 Aut n (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 \(\bar \Delta \) < Aut(T) has positive Hausdorff dimension. In other word

    $\mathop {\lim \inf }\limits_{n \to \infty } \frac{{\log (|\Psi _n (\Delta )|)}} {{\log (|Aut_n (T)|)}} > 0 $

    .

  • 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 \(Y_n = \mathcal{G}(\Delta ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{w} ,L_n )\) coming from the action of Δ on the different levels of the tree form a family of expander graphs.

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Correspondence to Yair Glasner.

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The author was partially supported by ISF grant 888/07 and BSF grant 2006-222.

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Glasner, Y. Strong approximation in random towers of graphs. Combinatorica 34, 139–172 (2014). https://doi.org/10.1007/s00493-014-2620-7

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Mathematics Subject Classification (2010)

  • 20E08, 60J80
  • 20B27, 28A78, 11F06