TY - JOUR

T1 - Strong approximation in random towers of graphs

AU - Glasner, Yair

N1 - Funding Information:
Acknowledgments. I thank Shlomo Hoory and Bálint Virág who pointed out that the proof of Bilu and Linial actually yields the precise statement required for this paper. I thank Miklós Abért for many stimulating conversations on random group actions, and for explaining to me some of the details of the paper [1]. Much of this paper was written in Geneva, I am very thankful to the Swiss science foundation that enabled that visit and to the Department of Mathematics at Geneva for their warm hospitality. This work is based on research done at the institute for advanced studies and supported by U.S. National Science Foundation under agreement DMS-0111298. The work was also supported by ISF grant 441/11 and BSF grant 2006-222.
Funding Information:
* The author was partially supported by ISF grant 888/07 and BSF grant 2006-222.

PY - 2014/1/1

Y1 - 2014/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84901988531&partnerID=8YFLogxK

U2 - 10.1007/s00493-014-2620-7

DO - 10.1007/s00493-014-2620-7

M3 - Article

AN - SCOPUS:84901988531

SN - 0209-9683

VL - 34

SP - 139

EP - 172

JO - Combinatorica

JF - Combinatorica

IS - 2

ER -