Expander Graphs and Amenable Quotients

  • Yehuda Shalom
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 109)

Abstract

We continue the search, carried out in [Sh1], for new sets of generators for families of finite groups (such as S L 2(F P )), which make the corresponding Cayley graphs an expander family. Along the way to our new result, we survey some of the recent results and methods introduced in [Sh1], based on the use of invariant means on the profinite completion of the finite groups.

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References

  1. [BS]
    M. Burger and P. Sarnak, Ramanujan duals II, Invent. Math. 106 (1991), 1–11.MathSciNetMATHCrossRefGoogle Scholar
  2. [DS]
    N. Dunford and J.T. Schwarts, LINEAR OPERATORS, part I, Interscience publishers, New-York, 1963.MATHGoogle Scholar
  3. [Ey]
    P. Eymard, Moyennes Invariants et Representations Unitairs, Lecture Notes in Mathematics 300, Springer Verlag 1972.Google Scholar
  4. [IN]
    A. Iozzi and A. Nevo, Algebraic hulls and the Folner property, to appear in GAFA.Google Scholar
  5. [Lub1]
    A. Lubotzky, DISCRETE GROUPS EXPANDING GRAPHS AND INVARIANT MEASURES, Birkhauser, 1994.Google Scholar
  6. [Lub2]
    A. Lubotzky, Cayley graphs: eigenvalues, expanders and random walks, in: Surveys in Combinatorics, 1995 (P. Rowbinson ed.), London Math. Soc. Lecture Note Ser. 218, Camb. Univ. Press, 1995, 155–189.Google Scholar
  7. [LPS1]
    A. Lubotzky, R. Phillips and P. Sarnak, Ramanujan graphs, Combinatorica 8 (1988), 261–277.MathSciNetMATHCrossRefGoogle Scholar
  8. [LPS2]
    A Lubotzky, R. Phillips and P. Sarnak, Hecke operators and distributing points on S 2, I, Comm. Pure and Applied Math. 39 (1986), S 149–186.MathSciNetCrossRefGoogle Scholar
  9. [LR1]
    J. Lafferty and D. Rockmore, Fast Fourier analysis for S L 2 over a finite field and related numerical experiments, Experimental Mathematics 1 (1992), 45–139.MathSciNetCrossRefGoogle Scholar
  10. [LR2]
    J. Lafferty and D. Rockmore, Level spacings for S L(2, p), this volume of IMA Volumes in Mathematics and its Applications (1997).Google Scholar
  11. [LRS]
    W. Luo, Z. Rudnick and P. Sarnak, On Selberg’s eigenvalue conjecture, GAFA 5 No. 2 (1995).Google Scholar
  12. [LW]
    A. Lubotzky and B. Weiss, Groups and expanders, in: “Expanding graphs” 95-109, DIMACS series Vol 10, American Math. Soc. 1993, (Ed. J. Friedman).Google Scholar
  13. [Mar1]
    G.A. Margulis, Explicit constructions of concentrators, Prob. of Inform. Transf. 10 (1975), 325–332.Google Scholar
  14. [Mar2]
    G.A. Margulis, Explicit group theoretic constructions of combinatorial schemes and their applications for the construction of expanders and concentrators, J. Prob. of Inform. Transf. 24 (1988), 39–46.MathSciNetMATHGoogle Scholar
  15. [Mor]
    M. Morgenstern, Existence and explicit construction of q + 1 regular Ramanujan graphs for every prime power q, J. of Comb. Th. B, to appear.Google Scholar
  16. [Pi]
    G. Pisier, Quadratic forms in unitary operators, preprint.Google Scholar
  17. [Ro]
    J. Rosenblatt, Uniqueness of invariant means for measure preserving transformations, Trans. AMS 265 (1981), 623–636.MathSciNetMATHCrossRefGoogle Scholar
  18. [Sar]
    P. Sarnak, SOME APPLICATIONS OF MODULAR FORMS, Cambridge Tracts in Mathematics 99, Camb. Univ. Press 1990.Google Scholar
  19. [Sc]
    K. Schmidt, Amenability, Kazhdan’s property T, strong ergodicity and invariant means for ergodic group actions, Ergodic Theory and Dynamical Systems 1 (1981) 223–236.MathSciNetMATHCrossRefGoogle Scholar
  20. [Sh1]
    Y. Shalom, Expander graphs and invariant means, To appear in Combinatorica.Google Scholar
  21. [Sh2]
    Y. Shalom, Hecke operators of group actions and weak containment of unitary representations, in preparation.Google Scholar
  22. [Sh3]
    Y. Shalom, Invariant measures for algebraic actions, Zariski dense subgroups and Kazhdan’s property (T), To appear in Trans. of AMS.Google Scholar
  23. [Zi]
    R.J. Zimmer, ERGODIC THEORY AND SEMISIMPLE GROUPS, Birkhauser, 1984.Google Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Yehuda Shalom
    • 1
  1. 1.Institute of MathematicsHebrew UniversityJerusalemIsrael

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