Expanding graphs and invariant means


All the known explicit constructions of expander families are essentially obtained by considering a sequence of finite index normal subgroupsN i ◃Γ, and taking the Cayley graphs of Γ/N i w.r. to the projection of aglobal finite set of generators of Γ. For many of these examples (e.g. Γ=SL 2ℤ, Γ/N i SL 2(\(\mathbb{F}_p \)) we present first constructions of new, different, sets of generators for the finite quotients, which make the Cayley graphs an expander family. An intrinsic connection between the expanding property and uniqueness of the Haar measure on an appropriate compact group, as an invariant mean, is established and used in the construction of such generators.

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  1. [1]

    M. B. Bekka: On uniqueness of invariant means, to appear in Proc. of AMS.

  2. [2]

    K. P. S. Bashkara Rao andM. Bhaskara Rao:Theory of Charges, Academic Press, 1983.

  3. [3]

    M. Burger andP. Sarnak: Ramanujan duals II,Invent. Math.,106 (1991), 1–11.

    Google Scholar 

  4. [4]

    R. Brooks: The spectral geometry of a tower of coverings,J. of Diff. Geom.,23 (1986), 97–107.

    Google Scholar 

  5. [5]

    R. Brooks: The spectral geometry of the Apollonian packing,Comm. Pure. Appl. Math.,4 (1985), 359–366.

    Google Scholar 

  6. [6]

    T. Coulhon andL. Saloff-Coste: Variétés riemanniennes isométrique á l'infini,Ref. Mat. Iber.,11 (1995) no. 3, 687–726.

    Google Scholar 

  7. [7]

    C. Chou: Ergodic group actions with non-unique invariant means,Proc. A.M.S.,100 (1987), 647–650.

    Google Scholar 

  8. [8]

    J. Hempel: Coverings of Dehn fillings of surface bundles,Topology and its applications,24 (1982), 157–170.

    Google Scholar 

  9. [9]

    A. Lubotzky:Discrete Groups, Expanding Graphs and Invariant Measures, Birkhauser, 1994.

  10. [10]

    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.

  11. [11]

    A. Lubotzky, R. Phillips andP. Sarnak: Ramanujan graphs,Combinatorica,8 (1988), 261–277.

    Google Scholar 

  12. [12]

    A. Lubotzky, R. Phillips andP. Sarnak: Hecke operators and distributing points onS 2,I,Comm. Pure and Applied Math.,39 (1986), 149–186.

    Google Scholar 

  13. [13]

    J. Lafferty andD. Rockmore: Fast Fourier analysis forSL 2, over a finite field and related numerical experiments,Experimental Mathematics,1 (1992), 45–139.

    Google Scholar 

  14. [14]

    J. Lafferty andD. Rockmore: Level spacings forSL(2,p), To appear in “Emerging applications of number theory”, IMA Volumes in Mathematics, ed: D. Hejhal et al.

  15. [15]

    W. Luo, Z. Rudnick andP. Sarnak: On Selberg's eigenvalue conjecture,GAFA,5, (1995), No. 2, 387–401.

    Google Scholar 

  16. [16]

    A. Lubotzky andB. Weiss: Groups and expanders, in: “Expanding graphs” 95–109, DIMACS series Vol 10, American Math. Soc. 1993, (Ed. J. Fiedman).

  17. [17]

    G. A. Margulis Explicit constructions of concentratorsProb. of Inform. Transf. 10 (1975), 325–332.

    Google Scholar 

  18. [18]

    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.

    Google Scholar 

  19. [19]

    G. A. Margulis:Discrete Subgroups of Semisimple Lie Groups, Springer-Verlag 1991.

  20. [20]

    M. Morgenstern: Existence and explicit construction ofq+1 regular Ramanujan graphs for every prime powerq, J. of Comb. Th. B,62 (1994), No. 1, 44–62.

    Google Scholar 

  21. [21]

    S. J. Patterson: The exponent of convergence of Poincare series,Monat. f. Math. 82 (1976) 297–315.

    Google Scholar 

  22. [22]

    S. J. Patterson: Lectures on measures on limit sets of Kleinian groups, In: Analytical and Geometrical Aspects of Hyperbolic Space, Ed. D.B.A. Epstein, London Math. Soc. Lecture Notes Series 111, Cambridge Univ. Press, 1987.

  23. [23]

    G. Pisier: Quadratic forms in unitary operators, preprint.

  24. [24]

    R. Rigley: A quadratic parabolic group,Proc. of the Camb. Phil. Soc.,77 (1975), 281–288.

    Google Scholar 

  25. [25]

    J. Rosenblatt: Uniqueness of invariant means for measure preserving transformations,Trans. AMS,265 (1981), 623–636.

    Google Scholar 

  26. [26]

    P. Sarnak: Some Applications of Modular Forms,Cambridge Tracts in Mathematics,99, Camb. Univ. Press 1990.

  27. [27]

    K. Schmidt: Amenability, Kazhdan's property (T), strong ergodicity and invariant means for ergodic group actions,Ergodic Th. and Dynamical Sys.,1, 223–236 (1981).

    Google Scholar 

  28. [28]

    Y. Shalom: Hecke operators of group actions and weak containment of unitary representations, in preparation.

  29. [29]

    D. Sullivan:Related aspects of positivity in: Collection: Aspects of mathematics and its applications, North-Holland Math. (1986) 747–779.

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Partially sponsored by the Edmund Landau Center for research in Mathematical Analysis, supported by the Minerva Foundation (Germany).

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Shalom, Y. Expanding graphs and invariant means. Combinatorica 17, 555–575 (1997). https://doi.org/10.1007/BF01195004

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Mathematics Subject Classification (1991)

  • 05C25
  • 20F32
  • 22E40
  • 28D15