Skip to main content
SpringerLink
Log in

Expanders, rank and graphs of groups

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

LetG be a finitely presented group, and let {G i } be a collection of finite index normal subgroups that is closed under intersections. Then, we prove that at least one of the following must hold: 1.G i is an amalgamated free product or HNN extension, for infinitely manyi; 2. the Cayley graphs ofG/G i (with respect to a fixed finite set of generators forG) form an expanding family; 3. infi(d(G i )−1)/[G:G i ]=0, whered(G i ) is the rank ofG i . The proof involves an analysis of the geometry and topology of finite Cayley graphs. Several applications of this result are given.

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. D. Cooper, D. Long and A. Reid,Essential closed surfaces in bounded 3-manifolds, Journal of the American Mathematical Society10 (1997), 553–563.

    Article  MATH  MathSciNet  Google Scholar 

  2. P. de la Harpe,Topics in Geometric Group Theory, Chicago Lectures in Mathematics, 2000.

  3. R. Grigorchuk,Burnside's problem on periodic groups, Functional Analysis and its Applications14 (1980), 41–43.

    MATH  MathSciNet  Google Scholar 

  4. M. Lackenby,Heegaard splittings, the virtually Haken conjecture and Property (τ), Preprint.

  5. M. Lackenby,A characterisation of large finitely presented groups, Journal of Algebra, to appear.

  6. A. Lubotzky,Dimension function for discrete groups, inProceedings of Groups, St. Andrews 1985, London Mathematical Society Lecture Note Series 121, Cambridge University Press, 1986, pp. 254–262.

  7. A. Lubotzky,Discrete Groups, Expanding Graphs and Invariant Measures, Progress in Mathematics 125, Birkhäuser, Boston, 1994.

    MATH  Google Scholar 

  8. A. Lubotzky and B. Weiss,Groups and expanders, inExpanding Graphs (Princeton, 1992), DIMACS Series in Discrete Mathematics and Theoretical Computer Science 10, American Mathematical Society, Providence, RI, 1993, pp. 95–109.

    Google Scholar 

  9. R. Lyndon and P. Schupp,Combinatorial Group Theory, Springer-Verlag, Berlin, 1977.

    MATH  Google Scholar 

  10. G. Margulis,Explicit construction of expanders, Problemy Peredav di Informacii9 (1973), 71–80.

    MathSciNet  Google Scholar 

  11. J-P. Serre,Le problème des groupes de congruence pour SL2, Annals of Mathematics92 (1970), 489–527.

    Article  MathSciNet  Google Scholar 

  12. J-P. Serre,Arbres, amalgames, SL2, Astérisque46 (1977).

  13. B. Sury and T. N. Venkataramana, Generators for all principal congruence subgroups of SL(n,ℤ) with n≥3, Proceedings of the American Mathematical Society122 (1994), 355–358.

    Article  MATH  MathSciNet  Google Scholar 

  14. W. Thurston,The Geometry and Topology of 3-Manifolds, Lecture Notes, Princeton, 1978.

Download references

Author information

Authors and Affiliations

  1. Mathematical Institute, Oxford University, 24-29 St Giles', OX1 3LB, Oxford, UK

    Marc Lackenby

Authors
  1. Marc Lackenby
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marc Lackenby.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lackenby, M. Expanders, rank and graphs of groups. Isr. J. Math. 146, 357–370 (2005). https://doi.org/10.1007/BF02773541

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02773541

Keywords

Search

Navigation