Can one hear the shape of a group?

  • Alain Valette


Let Γ be a finitely generated group, and letS be a finite, non-necessarily symmetric, generating subset of Γ. Leth be the transition operator of the directed Cayley graphG(Γ,S), acting onl2 (Γ). Staring with Kesten’s seminal results, we give a survey of results linking group-theoretic properties of the pair (Γ,S) with spectral properties ofh.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Car]Cartwright, D.I.,Singularities of the Green function of a random walk on a discrete group, Mh. Math.,113 (1992), 183–188.MATHCrossRefMathSciNetGoogle Scholar
  2. [Day]Day, M.M.,Convolutions, means and spectra, Illinois J. Math.,8 (1964), 100–111.MATHMathSciNetGoogle Scholar
  3. [Ell]Elliott, G.,Gaps in the spectrum of an almost periodic Schrödinger operator, C.R. Math. Ref. Acad. Sci. Canada,4 (1982), 255–259.MATHGoogle Scholar
  4. [GhH]Ghys, E. andde la Harpe, P.,Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Math.,83, Birkhäuser, 1990.Google Scholar
  5. [GWW]Gordon, C., Webb, D. andWolpert, S.,Isospectral plane domains and surfaces via Riemannian orbifolds, Invent. Math.,110 (1992), 1–22.MATHCrossRefMathSciNetGoogle Scholar
  6. [Gro]Gromov, M.,Hyperbolic groups, in Essays in Group Theory (S.M. Gersten Ed.), Springer, Berlin, 1987, pp. 75–263.Google Scholar
  7. [Har]de la Harpe, P.,Groupes hyperboliques, algèbres d’opérateurs, et un théoreme de Jolissaint, C.R. Acad. Sci. Paris (Sér. I),307 (1988), 771–774.MATHGoogle Scholar
  8. [HRV1]de la Harpe, P. andRobertson, A.G. andValette, A.,On the spectrum of the sum of generators for a finitely generated group, Israel J. Math.,81 (1993), 65–96.MATHCrossRefMathSciNetGoogle Scholar
  9. [HRV2]de la Harpe, P., Robertson, A.G. andValette, A.,On the spectrum of the sum of generators for a finitely generated group II, Coll. Math.,LXV, (1993), 87–102.Google Scholar
  10. [Hof]Hofstadter, D.,Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Phys. Review,B14 (1976), 2239–2249.Google Scholar
  11. [Jol]Jolissaint, P.,Rapidly decreasing functions in reduced C *-algebras of groups, Trans. Amer. Math. Soc.,317 (1990), 167–196.MATHCrossRefMathSciNetGoogle Scholar
  12. [Kac]Kac, M.,Can one hear the shape of a drum?, Amer. Math. Monthly,73 (1966), 1–23.CrossRefGoogle Scholar
  13. [KaT]Kaniuth, E. andTaylor, K.F.,Projections in C *-algebras of nilpotent groups, Manuscripta Math.,65 (1989), 93–111.MATHCrossRefMathSciNetGoogle Scholar
  14. [Ke1]Kesten, H.,Symmetric random walks on groups, Trans. Amer. Math. Soc.,92 (1959), 336–354.MATHCrossRefMathSciNetGoogle Scholar
  15. [Ke2]Kesten, H.,Full Banach mean values on countable groups, Math. Scand.,7 (1959), 146–156.MATHMathSciNetGoogle Scholar
  16. [PiV]Pimsner, M. andVoiculescu, D.,K-groups of reduced crossed products by free groups, J. Operator Theory,8 (1982), 131–156.MATHMathSciNetGoogle Scholar
  17. [RaB]Rammal, R. andBellissard, J.,An algebraic semiclassical approach to Bloch electrons in a magnetic field, J. Phys. France,51 (1990), 1803–1830.CrossRefGoogle Scholar
  18. [Rie]Rieffel, M.,C *-algebras associated with irrational rotations, Pacific J. Math.,95 (1981), 415–429.MathSciNetGoogle Scholar
  19. [Ros]Rosenblatt, J.M.,Invariant measures and growth conditions, Trans. Amer. Math. Soc.,193 (1974), 33–53.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser-Verlag 1994

Authors and Affiliations

  • Alain Valette
    • 1
  1. 1.Institut de MathématiquesNeuchâtelSwitzerland

Personalised recommendations