Mathematische Annalen

, Volume 354, Issue 1, pp 43–72 | Cite as

Spectral distribution and L2-isoperimetric profile of Laplace operators on groups

  • Alexander Bendikov
  • Christophe Pittet
  • Roman Sauer
Article

Abstract

We give a formula relating the L2-isoperimetric profile to the spectral distribution of a Laplace operator on a finitely generated group Γ. We prove the asymptotic stability of the spectral distribution under changes of measures with finite second moment. As a consequence, we can apply techniques from geometric group theory to estimate the spectral distribution of the Laplace operator in terms of the growth and the Følner’s function of the group. This leads to upper bounds on spectral distributions of some non-solvable amenable groups and to sharp estimates of the spectral distributions of some solvable groups with exponential growth.

Mathematics Subject Classification (2000)

Primary 58J35 20F63 Secondary 60G50 

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References

  1. 1.
    Bartholdi L., Woess W.: Spectral computations on lamplighter groups and Diestel-Leader graphs. J. Fourier Anal. Appl. 11(2), 175–202 (2005)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bendikov, A., Pittet, C., Sauer, R.: QI-invariance of the spectral distribution of Laplace operators and the return probability of random walks on groups. PreprintGoogle Scholar
  3. 3.
    Bendikov A., Coulhon T., Saloff-Coste L.: Ultracontractivity and embedding into L. Math. Ann. 337(4), 817–853 (2007)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bingham N.H., Goldie C.M., Teugels J.L.: Regular Variation. Encyclopedia of Mathematics and Its Applications, vol. 27. Cambridge University Press, Cambridge (1989)Google Scholar
  5. 5.
    Brown K.S.: Cohomology of Groups. Graduate Texts in Mathematics, vol. 87. Springer, Berlin (1982)Google Scholar
  6. 6.
    Varopoulos N.Th., Saloff-Coste L., Coulhon T.: Analysis and Geometry on Groups. Cambridge Tracts in Mathematics, vol. 100. Cambridge University Press, Cambridge (1992)Google Scholar
  7. 7.
    Coulhon, T.: Heat Kernel and Isoperimetry on Non-Compact Riemannian Manifolds (Paris, 2002). Contemporary Mathematics, vol. 338, pp. 65–99. American Mathematical Society, Providence (2003)Google Scholar
  8. 8.
    Coulhon T., Saloff-Coste L.: Isopérimétrie pour les groupes et les variétés. Rev. Mat. Iberoamericana 9(2), 293–314 (1993) (French)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Coulhon T., Saloff-Coste L.: Variétés riemanniennes isométriques à l’infini. Rev. Mat. Iberoamericana 11(3), 687–726 (1995) (French)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Coulhon T., Grigor’yan A., Pittet C.: A geometric approach to on-diagonal heat kernel lower bounds on groups. Ann. Inst. Fourier (Grenoble) 51(6), 1763–1827 (2001) (English, with English and French summaries)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Coulhon T., Grigor’yan A.: On-diagonal lower bounds for heat kernels and Markov chains. Duke Math. J. 89(1), 133–199 (1997)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Dicks W., Schick T.: The spectral measure of certain elements of the complex group ring of a wreath product. Geom. Dedicata 93, 121–137 (2002)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Dodziuk J.: Difference equations, isoperimetric inequality and transience of certain random walks. Trans. Am. Math. Soc. 284(2), 787–794 (1984)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Efremov, D.V., Shubin, M.A.: Spectrum distribution function and variational principle for auto- morphic operators on hyperbolic space. Séminaire sur les Équations aux Dérivées Partielles, 1988–1989, É cole Polytech., Palaiseau (1989)Google Scholar
  15. 15.
    Erschler A.: Isoperimetry for wreath products of Markov chains and multiplicity of selfintersections of random walks. Probab. Theory Relat. Fields 136(4), 560–586 (2006)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Erschler A.: Piecewise automatic groups. Duke Math. J. 134(3), 591–613 (2006)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Følner E.: On groups with full Banach mean value. Math. Scand. 3, 243–254 (1955)MathSciNetGoogle Scholar
  18. 18.
    Gretete D.: Stabilité du comportement des marches aléatoires sur un groupe localement compact. Ann. Inst. Henri Poincaré Probab. Stat. 44(1), 129–142 (2008) (French, with English and French summaries)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Grigorchuk R.I., Linnell P., Schick T.,  Zuk A.: On a question of Atiyah. C. R. Acad. Sci. Paris Sér. I Math. 331(9), 663–668 (2000) (English, with English and French summaries)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Gromov M.: Asymptotic Invariants of Infinite Groups. Geometric Group Theory, vol. 2 (Sussex, 1991). London Mathematical Society Lecture Note Series, vol. 182, pp. 1–295. Cambridge University Press, Cambridge (1993)Google Scholar
  21. 21.
    Gromov M.: Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. 53, 53–73 (1981)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Gromov M.: Entropy and isoperimetry for linear and non-linear group actions. Groups Geom Dyn 1(4), 499–593 (2008)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Gromov M., Shubin M.: von Neumann spectra near zero. Geom. Funct. Anal. 1(4), 375–404 (1991)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Kaĭmanovich V.A., Vershik A.M.: Random walks on discrete groups: boundary and entropy. Ann. Probab. 11(3), 457–490 (1983)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Kaĭmanovic˘, V.A.: The spectral measure of transition operator and harmonic functions connected with random walks on discrete groups. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 97, 102–109, 228–229, 236 (1980) (Russian, with English summary). Problems of the theory of probability distributions, VIGoogle Scholar
  26. 26.
    Kesten H.: Full Banach mean values on countable groups. Math. Scand. 7, 146–156 (1959)MathSciNetMATHGoogle Scholar
  27. 27.
    Lück W.: L2-Invariants: Theory and Applications to Geometry and K-Theory. Springer, Berlin (2002)Google Scholar
  28. 28.
    Mohar B., Woess W.: A survey on spectra of infinite graphs. Bull. Lond. Math. Soc. 21(3), 209–234 (1989)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Müller P., Stollmann P.: Spectral asymptotics of the Laplacian on supercritical bond-percolation graphs. J. Funct. Anal. 252(1), 233–246 (2007)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Oguni, S.-I.: Spectral density functions of general modules over finite von Neumann algebras and their applications. PreprintGoogle Scholar
  31. 31.
    Pittet C.: The isoperimetric profile of homogeneous Riemannian manifolds. J. Differ. Geom. 54(2), 255–302 (2000)MathSciNetMATHGoogle Scholar
  32. 32.
    Pittet C., Saloff-Coste L.: Amenable Groups, Isoperimetric Profiles and Random Walks. Geometric Group Theory Down Under (Canberra, 1996), pp. 293–316. de Gruyter, Berlin (1999)Google Scholar
  33. 33.
    Pittet C., Saloff-Coste L.: On random walks on wreath products. Ann. Probab. 30(2), 948–977 (2002)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Pittet C., Saloff-Coste L.: Random walks on finite rank solvable groups. J. Eur. Math. Soc. (JEMS) 5(4), 313–342 (2003)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Pittet C., Saloff-Coste L.: On the stability of the behavior of random walks on groups. J. Geom. Anal. 10(4), 713–737 (2000)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Riesz F., Sz.-Nagy B.: Leçons d’Analyse Fonctionnelle. Gauthier-Villars, Paris (1972)Google Scholar
  37. 37.
    Rockafellar, R.T.: Convex Analysis. Princeton Landmarks in Mathematics. Princeton University Press, Princeton (1997). Reprint of the 1970 original; Princeton PaperbacksGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Alexander Bendikov
    • 1
  • Christophe Pittet
    • 2
    • 3
  • Roman Sauer
    • 4
    • 5
  1. 1.Institute of MathematicsWroclaw UniversityWrocławPoland
  2. 2.CMI Université d’Aix-Marseille IMarseilleFrance
  3. 3.Laboratoire Poncelet CNRSMoscowRussia
  4. 4.Department of MathematicsUniversity of ChicagoChicagoUSA
  5. 5.Fakultät für MathematikUniversität RegensburgRegensburgGermany

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