Israel Journal of Mathematics

, Volume 81, Issue 1–2, pp 65–96 | Cite as

On the spectrum of the sum of generators for a finitely generated group

  • Pierre de la Harpe
  • A. Guyan Robertson
  • Alain Valette


Let Γ be a finitely generated group. In the group algebra ℂ[Γ], form the averageh of a finite setS of generators of Γ. Given a unitary representation π of Γ, we relate spectral properties of the operator π(h) to properties of Γ and π.

For the universal representationπ un of Γ, we prove in particular the following results. First, the spectrum Sp(π un (h)) contains the complex numberz of modulus one iff Sp(π un (h)) is invariant under multiplication byz, iff there exists a character\(\chi :\Gamma \to \mathbb{T}\) such that η(S)={z}. Second, forS −1=S, the group Γ has Kazhdan’s property (T) if and only if 1 is isolated in Sp(π un (h)); in this case, the distance between 1 and other points of the spectrum gives a lower bound on the Kazhdan constants. Numerous examples illustrate the results.


Hilbert Space Irreducible Representation Unitary Representation Cayley Graph Free Product 
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  1. [AIM] N. Alon and V.D. Milman, λ1,Isoperimetric inequalities for graphs and super-concentrators, J. Combinatorial Theory Ser.B38 (1985), 73–88.CrossRefMathSciNetGoogle Scholar
  2. [BaH] R. Bacher and P. de la Harpe,Exact values of Kazhdan constants for some finite groups, preprint, Geneva, 1991.Google Scholar
  3. [BeH] E. Bedos and P. de la Harpe,Moyennabilité intérieure des groupes: définitions et examples, L’Enseignement Math.32 (1986), 139–157.MATHGoogle Scholar
  4. [Bek] M. E. B. Bekka,Amenable unitary representations of locally compact groups, Invent. Math.100 (1990), 383–401.MATHCrossRefMathSciNetGoogle Scholar
  5. [Bie] F. Bien,Constructions of telephone networks by group representations, Notices Amer. Math. Soc.,36–1 (1989), 5–22.MathSciNetGoogle Scholar
  6. [BMS] N. L. Biggs, B. Mohar and J. Shawe-Taylor,The spectral radius of infinite graphs, Bull. London Math. Soc.20 (1988), 116–120.MATHCrossRefMathSciNetGoogle Scholar
  7. [Bur] M. Burger,Kazhdan constants for SL(3, ℤ, J. Reine Angew. Math.43 (1991), 36–67.MathSciNetGoogle Scholar
  8. [Car] D. I. Cartwright,Singularities of the Green function of a random walk on a discrete group, Mh. Math.,113 (1992), 183–188.MATHCrossRefMathSciNetGoogle Scholar
  9. [CS1] D. I. Cartwright and P. M. Soardi,Harmonic analysis on the free product of two cyclic groups, J. Funct. Anal.65 (1986), 147–171.MATHCrossRefMathSciNetGoogle Scholar
  10. [CS2] D. I. Cartwright and P. M. Soardi,Random walks on free products, quotients and amalgams, Nagoya Math. J.102 (1986), 163–180.MATHMathSciNetGoogle Scholar
  11. [CdV] Y. Colin de Verdière,Distribution de Points sur une Sphère (d’après Lubotzky, Phillips et Sarnak), Sem. Bourbaki 703, 1988.Google Scholar
  12. [DeK] C. Delaroche and A. Kirillov,Sur les relations entre l’espace dual d’un groupe et la structure de ses sous-groupes fermés, Sem. Bourbaki343 1967–1968.Google Scholar
  13. [Dix] J. Dixmier,C *-algebras, North-Holland, Amsterdam-New York-Oxford, 1977.MATHGoogle Scholar
  14. [Fel] J. M. G. Fell,Weak containment and Kronecker products of group representations, Pacific. J. Math.13 (1963), 503–510.MATHMathSciNetGoogle Scholar
  15. [Gan] F. Gantmacher,The Theory of Matrices, Vol. 2, Chelsea Publ., New York, 1960.Google Scholar
  16. [Ger] P. Gerl,Amenable groups and amenable graphs, Lecture Notes in Math. 1359, Springer, Berlin, 1988, pp. 181–190.Google Scholar
  17. [GoM] C. D. Godsil and B. Mohar,Walk generating functions and spectral measures of infinite graphs, Linear Algebra and its Appl.107 (1988), 191–206.MATHCrossRefMathSciNetGoogle Scholar
  18. [Gra] J. E. Graver and M. E. Watkins,Combinatorics with Emphasis on the Theory of Graphs, Springer-Verlag, New York, 1977.MATHGoogle Scholar
  19. [Gro] M. Gromov,Hyperbolic groups, inEssays in Group Theory (S. M. Gersten, ed.), Springer, Berlin, 1987, pp. 75–263.Google Scholar
  20. [Hal] P. R. Halmos,A Hilbert Space Problem Book (2nd ed.), Springer-Verlag, New York, 1982.MATHGoogle Scholar
  21. [HaV] P. de la Harpe and A. Valette,La propriété (T) de Kazhdan pour les groupes localement compacts, Astérique175, Soc. Math. France, 1989.Google Scholar
  22. [HRV] P. de la Harpe, A. G. Robertson and A. Valette,On exactness of group C *-algebras, preprint, 1991.Google Scholar
  23. [Hul] A. Hulanicki,Groups whose regular representation weakly contains all unitary representations, Studia Math.24 (1964), 37–59. See also:Means and Følner conditions on locally compact groups, Studia Math.27 (1966), 87–104.MathSciNetGoogle Scholar
  24. [IoP] A. Iozzi and M. A. Picardello,Spherical functions on symmetric graphs, inConference in Harmonic Analysis, Lecture Notes in Math. 992, Springer, Berlin, 1983, pp. 87–104.CrossRefGoogle Scholar
  25. [Kaz] D. Kazhdan,Connection of the dual space of a group with the structure of its closed subgroups, Funct. Anal. Appl.1 (1967), 63–65.MATHCrossRefGoogle Scholar
  26. [Ke1] H. Kesten,Symmetric random walks on groups, Trans. Amer. Math. Soc.92 (1959), 336–354.MATHCrossRefMathSciNetGoogle Scholar
  27. [Ke2] H. Kesten,Full Banach mean values on countable groups, Math. Scand.7 (1959), 146–156.MATHMathSciNetGoogle Scholar
  28. [KuS] G. Kuhn and P. M. Soardi,The Plancherel measure for polygonal graphs, Ann. Mat. Pura. Appl.134 (1983), 393–401.MATHCrossRefMathSciNetGoogle Scholar
  29. [Lub] A. Lubotzky,Discrete groups, expanding graphs and invariant measures, CBMS Proceedings (1989) Amer. Math. Soc. (to appear).Google Scholar
  30. [Mlo] W. Mlotkowski,Positive definite functions on free products of groups, Boll. Un. Mat. Ital.3-B (1989), 343–355.MathSciNetGoogle Scholar
  31. [MoW] B. Mohar and W. Woess,A survey on spectra of infinite graphs, Bull. London Math. Soc.21 (1989), 209–234.MATHCrossRefMathSciNetGoogle Scholar
  32. [Pat] A. L. T. Paterson,Amenability, Math. Surveys and Monographs 29, Amer. Math. Soc., 1988.Google Scholar
  33. [Ped] G. K. Pederson,C *-algebras and their Automorphism Groups, Academic Press, London-New York-San Francisco, 1979.Google Scholar
  34. [RSN] F. Riesz and B. Sz-Nagy,Leçons d’analyse fonctionelle, Akadémiai Kiadó, Budapest, 1952.Google Scholar
  35. [SoW] P. M. Soardi and W. Woess,Amenability, unimodularity, and the spectral radius of random walks on infinite graphs, preprint, Milano, 1988.Google Scholar
  36. [Ste] R. Steinberg,Some consequences of the elementary relations in SLn, Contemporary Math.45 (1985), 335–350.MathSciNetGoogle Scholar
  37. [Tit] J. Tits,Free subgroups in linear groups, J. Algebra20 (1972), 250–270.MATHCrossRefMathSciNetGoogle Scholar
  38. [Val] A. Valette,Minimal projections, integrable representations and property (T), Arch. Math.43 (1984), 397–406.MATHCrossRefMathSciNetGoogle Scholar
  39. [Wan] S. P. Wang,On isolated points in the dual space of locally compact groups, Math. Ann.218 (1975), 19–34.MATHCrossRefMathSciNetGoogle Scholar
  40. [Was] S. Wassermann,C *-algebras associated with groups with Kazhdan’s property T, Ann. Math.134 (1991), 423–431.CrossRefMathSciNetGoogle Scholar
  41. [Wyn] A. D. Wyner,Random packings and coverings of the unit n-sphere, Bell System Tech. J.46 (1967), 2111–2118.MathSciNetGoogle Scholar
  42. [Yos] H. Yoshizawa,Some remarks on unitary representations of the free group, Osaka Math. J.3 (1951), 55–63.MATHMathSciNetGoogle Scholar

Copyright information

© Hebrew University 1993

Authors and Affiliations

  • Pierre de la Harpe
    • 1
  • A. Guyan Robertson
    • 2
  • Alain Valette
    • 3
  1. 1.Section de mathématiquesGenève 24Switzerland
  2. 2.Mathematics DepartmentUniversity of NewcastleAustralia
  3. 3.Institut de mathématiquesNeuchâtelSwitzerland

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