Skip to main content
SpringerLink
Log in

Strong approximation in random towers of graphs

  • Original Paper
  • Published:
Combinatorica Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Miklós Abért and Bálint Virág: Dimension and randomness in groups acting on rooted trees, J. Amer. Math. Soc. 18 (2005), 157–192 (electronic).

    Article  MATH  MathSciNet  Google Scholar 

  2. Jean Bourgain and Alex Gamburd: Expansion and random walks in SLd(ℤ/p nℤ), I, J. Eur. Math. Soc. (JEMS) 10 (2008), 987–1011.

    Article  MATH  MathSciNet  Google Scholar 

  3. Jean Bourgain and Alex Gamburd: Random walks and expansion inSLd(ℤ/p nℤ), C. R. Math. Acad. Sci. Paris 346 (2008), 619–623.

    Article  MATH  MathSciNet  Google Scholar 

  4. Jean Bourgain and Alex Gamburd: Uniform expansion bounds for Cayley graphs of \(SL_2 (\mathbb{F}_p )\), Ann. of Math. (2) 167 (2008), 625–642.

    Article  MATH  MathSciNet  Google Scholar 

  5. Jean Bourgain and Alex Gamburd: Expansion and random walks in SLd(ℤ/p nℤ), II, J. Eur. Math. Soc. (JEMS) 11 (2009), 1057–1103; with an appendix by Bourgain.

    Article  MATH  MathSciNet  Google Scholar 

  6. Emmanuel Breuillard, Ben Green, Robert Guralnick and Terence Tao: Expansion in finite simple groups of lie type, preprint, 2011.

  7. Emmanuel Breuillard, Ben Green, Robert Guralnick and Terence Tao: Strongly dense free subgroups of semisimple algebraic groups, Israel J. Math. 192 (2012), 347–379.

    Article  MATH  MathSciNet  Google Scholar 

  8. Meenaxi Bhattacharjee: The ubiquity of free subgroups in certain inverse limits of groups, J. Algebra 172 (1995), 134–146.

    Article  MATH  MathSciNet  Google Scholar 

  9. Y. Barnea and M. Larsen: Random generation in semisimple algebraic groups over local fields, J. Algebra 271 (2004), 1–10.

    Article  MATH  MathSciNet  Google Scholar 

  10. Yonatan Bilu and Nathan Linial: Lifts, discrepancy and nearly optimal spectral gap, Combinatorica 26 (2006), 495–519.

    Article  MATH  MathSciNet  Google Scholar 

  11. Marc Burger and Shahar Mozes: Groups acting on trees: from local to global structure, Inst. Hautes é Etudes Sci. Publ. Math. 92 (2002) 113–150.

    Article  MathSciNet  Google Scholar 

  12. Yiftach Barnea and Aner Sahlev: Hausdorff dimension, pro-p groups, and Kac-Moody algebras, Trans. Amer. Math. Soc. 349 (1997), 5073–5091.

    Article  MATH  MathSciNet  Google Scholar 

  13. Jean Bourgain and Péter P. Varjéu: Expansion in SLd(ℤ/qℤ); q arbitrary, Invent. Math. 188 (2012), 151–173.

    Article  MATH  MathSciNet  Google Scholar 

  14. John D. Dixon: The probability of generating the symmetric group, Math. Z., 110 (1969), 199–205.

    Article  MATH  MathSciNet  Google Scholar 

  15. Alex Gamburd: On the spectral gap for infinite index “congruence” subgroups of SL2(Z), Israel J. Math., 127 (2002), 157–200.

    Article  MathSciNet  Google Scholar 

  16. H. A. Helfgott: Growth and generation in SL2(ℤ/pℤ). Ann. of Math. (2) 167 (2008), 601–623.

    Article  MATH  MathSciNet  Google Scholar 

  17. Ehud Hrushovski and Anand Pillay: Definable subgroups of algebraic groups over finite fields, J. Reine Angew. Math. 462 (1995), 69–91.

    MATH  MathSciNet  Google Scholar 

  18. William M. Kantor and Alexander Lubotzky: The probability of generating a finite classical group, Geom. Dedicata 36 (1990), 67–87.

    Article  MATH  MathSciNet  Google Scholar 

  19. Martin W. Liebeck and Aner Shalev: The probability of generating a finite simple group, Geom. Dedicata 56 (1995), 103–113.

    Article  MATH  MathSciNet  Google Scholar 

  20. Alexander Lubotzky and Dan Segal: Subgroup growth, volume 212 of Progress in Mathematics, Birkhäuser Verlag, Basel, 2003.

    Book  Google Scholar 

  21. Alexander Lubotzky: Expander graphs in pure and applied mathematics, Bull. Amer. Math. Soc. (N.S.) 49 (2012), 113–162.

    Article  MATH  MathSciNet  Google Scholar 

  22. Nikolay Nikolov: Strong approximation methods, in: Lectures on pro-finite topics in group theory, volume 77 of London Math. Soc. Stud. Texts, 63–97, Cambridge Univ. Press, Cambridge, 2011.

    Google Scholar 

  23. Madhav V. Nori: On subgroups of \(GL_n (\mathbb{F}_p )\), Invent. Math. 88 (1987), 257–275.

    Article  MATH  MathSciNet  Google Scholar 

  24. Richard Pink: Strong approximation for Zariski dense subgroups over arbitrary global fields, Comment. Math. Helv. 75 (2000), 608–643.

    Article  MATH  MathSciNet  Google Scholar 

  25. Vladimir Platonov and Andrei Rapinchuk: Algebraic groups and number theory, volume 139 of Pure and Applied Mathematics, Academic Press Inc., Boston, MA, 1994; translated from the 1991 Russian original by Rachel Rowen.

    Google Scholar 

  26. Yehuda Shalom: Expanding graphs and invariant means, Combinatorica 17 (1997), 555–575.

    Article  MATH  MathSciNet  Google Scholar 

  27. Yehuda Shalom: Expander graphs and amenable quotients, in: Emerging applications of number theory (Minneapolis, MN, 1996), volume 109 of IMA Vol. Math. Appl., 571–581, Springer, New York, 1999.

    Chapter  Google Scholar 

  28. Boris Weisfeiler: Strong approximation for Zariski-dense subgroups of semisimple algebraic groups, Ann. of Math. (2) 120 (1984), 271–315.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Ben-Gurion University of the Negev, Be’er Sheva, Israel

    Yair Glasner

Authors
  1. Yair Glasner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yair Glasner.

Additional information

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

Rights and permissions

Reprints and permissions

About this article

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00493-014-2620-7

Mathematics Subject Classification (2010)

Search

Navigation