Israel Journal of Mathematics

, Volume 146, Issue 1, pp 357–370

Expanders, rank and graphs of groups



LetG be a finitely presented group, and let {Gi} 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.Gi is an amalgamated free product or HNN extension, for infinitely manyi; 2. the Cayley graphs ofG/Gi (with respect to a fixed finite set of generators forG) form an expanding family; 3. infi(d(Gi)−1)/[G:Gi]=0, whered(Gi) is the rank ofGi. The proof involves an analysis of the geometry and topology of finite Cayley graphs. Several applications of this result are given.


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  1. [1]
    D. Cooper, D. Long and A. Reid,Essential closed surfaces in bounded 3-manifolds, Journal of the American Mathematical Society10 (1997), 553–563.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    P. de la Harpe,Topics in Geometric Group Theory, Chicago Lectures in Mathematics, 2000.Google Scholar
  3. [3]
    R. Grigorchuk,Burnside's problem on periodic groups, Functional Analysis and its Applications14 (1980), 41–43.MATHMathSciNetGoogle Scholar
  4. [4]
    M. Lackenby,Heegaard splittings, the virtually Haken conjecture and Property (τ), Preprint.Google Scholar
  5. [5]
    M. Lackenby,A characterisation of large finitely presented groups, Journal of Algebra, to appear.Google Scholar
  6. [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.Google Scholar
  7. [7]
    A. Lubotzky,Discrete Groups, Expanding Graphs and Invariant Measures, Progress in Mathematics 125, Birkhäuser, Boston, 1994.MATHGoogle Scholar
  8. [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. [9]
    R. Lyndon and P. Schupp,Combinatorial Group Theory, Springer-Verlag, Berlin, 1977.MATHGoogle Scholar
  10. [10]
    G. Margulis,Explicit construction of expanders, Problemy Peredav di Informacii9 (1973), 71–80.MathSciNetGoogle Scholar
  11. [11]
    J-P. Serre,Le problème des groupes de congruence pour SL2, Annals of Mathematics92 (1970), 489–527.CrossRefMathSciNetGoogle Scholar
  12. [12]
    J-P. Serre,Arbres, amalgames, SL2, Astérisque46 (1977).Google Scholar
  13. [13]
    B. Sury and T. N. Venkataramana, Generators for all principal congruence subgroups of SL(n,) with n≥3, Proceedings of the American Mathematical Society122 (1994), 355–358.MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    W. Thurston,The Geometry and Topology of 3-Manifolds, Lecture Notes, Princeton, 1978.Google Scholar

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© The Hebrew University Magnes Press 2005

Authors and Affiliations

  1. 1.Mathematical InstituteOxford UniversityOxfordUK

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