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Critical exponents of discrete groups and $L^2$-spectrum


Author: Enrico Leuzinger
Journal: Proc. Amer. Math. Soc. 132 (2004), 919-927
MSC (2000): Primary 22E40, 53C20, 53C35
DOI: https://doi.org/10.1090/S0002-9939-03-07173-9
Published electronically: September 12, 2003
MathSciNet review: 2019974
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Abstract: Let $G$ be a noncompact semisimple Lie group and $\Gamma$ an arbitrary discrete, torsion-free subgroup of $G$. Let $\lambda_0(M)$ be the bottom of the spectrum of the Laplace-Beltrami operator on the locally symmetric space $M=\Gamma\backslash X$, and let $\delta(\Gamma)$ be the exponent of growth of $\Gamma$. If $G$ has rank $1$, then these quantities are related by a well-known formula due to Elstrodt, Patterson, Sullivan and Corlette. In this note we generalize that relation to the higher rank case by estimating $\lambda_0(M)$ from above and below by quadratic polynomials in $\delta(\Gamma)$. As an application we prove a rigiditiy property of lattices.


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  • 1. P.ALBUQUERQUE, Patterson-Sullivan theory in higher rank symmetric spaces, Geom. Funct. Anal. 9 (1999), 1-28. MR 2000d:37021
  • 2. J.-P.ANKER AND L.JI, Heat kernel and Green function estimates on noncompact symmetric spaces, Geom. Funct. Anal. 9 (1999), 1035-1091. MR 2001b:58038
  • 3. Y.BENOIST, Propriétés asymptotiques des groupes linéaires, Geom. Funct. Anal. 7 (1997), 1-47. MR 98b:22010
  • 4. N.BOURBAKI, Éléments de mathématique, Groupes et Algèbres de Lie, Chapitres IV, V, VI, Paris, 1981. MR 83g:17001
  • 5. I.CHAVEL, Eigenvalues in Riemannian Geometry, Pure and Applied Mathematics, Vol. 115, Academic Press, Orlando, FL, 1984. MR 86g:58140
  • 6. J.-P. CONZE AND Y. GUIVARC'H, Densité d'orbites d'actions de groupes linéaires, in: Rigidity in Dynamics and Geometry, M. Burger and A. Iozzi (eds.), Springer-Verlag, Berlin, 2002. MR 2003b:00029
  • 7. K.CORLETTE, Hausdorff dimensions of limit sets. I, Invent. Math. 102 (1990), 521-541. MR 91k:58067
  • 8. J.ELSTRODT, Die Resolvente zum Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. Teil I. Math. Ann. 203 (1973), 295-330, Teil II. Math. Z. 132 (1973), 99-134, Teil III. Math. Ann. 208 (1974), 99-132. MR 50:12922a, MR 50:12922b, MR 50:12922c
  • 9. Y. GUIVARC'H, Produits de matrices aléatoires et applications aux propriétés géométriques des sous-groupes du groupe linéaire, Ergodic Theory Dynam. Systems 10 (1990), 483-512. MR 92c:60011
  • 10. S. HELGASON, Differential Geometry, Lie Groups, and Symmetric Spaces, Pure and Applied Mathematics, Vol. 80, Academic Press, New York, 1978. MR 80k:53081
  • 11. G.KNIEPER, On the asymptotic geometry of nonpositively curved manifolds, Geom. Funct. Anal. 7 (1997) 755-782. MR 98h:53055
  • 12. V. G. KAC AND E. B. VINBERG, Quasi-homogeneous cones, Math. Notes 1 (1967), 231-235; translated from Mat. Zametki 1 (1967), 347-354. MR 34:8280
  • 13. E. LEUZINGER, Kazhdan's property (T), $L^2$-spectrum and isoperimetric inequalities for locally symmetric spaces, Comment. Math. Helv. 78 (2003), 116-133.
  • 14. A.LUBOTZKY, S.MOZES, AND M. S RAGHUNATHAN, The word and Riemannian metrics on lattices of semisimple groups, Inst. Hautes Études Sci. Publ. Math. 91 (2000), 5-53. MR 2002e:22011
  • 15. G. A.MARGULIS, Discrete Subgroups of Semisimple Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 17, Springer-Verlag, New York, 1991. MR 92h:22021
  • 16. G. A.MARGULIS AND G. A.SOIFER, Maximal subgroups of infinite index in finitely generated linear groups, J. of Algebra 69 (1981), 1-23. MR 83a:20056
  • 17. G. D.MOSTOW, Strong Rigidity of Locally Symmetric Spaces, Ann. Math. Stud. 78 (1978). MR 52:5874
  • 18. S.PATTERSON, The limit set of a Fuchsian group, Acta Math. 136 (1976), 241-273. MR 56:8841
  • 19. J.-F.QUINT, Mesures de Patterson-Sullivan en rang supérieur, Geom. Funct. Anal. 12 (2002), 776-809.
  • 20. J.-F.QUINT, Divergence exponentielle des sous-groupes discrets en rang supérieur, Comment. Math. Helv. 77 (2002), 563-608. MR 2003j:22015
  • 21. R.STRICHARTZ, Analysis of the Laplacian on the complete Riemannian manifold, J. Funct. Anal. 52 (1983), 48-79. MR 84m:58138
  • 22. D.SULLIVAN, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 171-202. MR 81b:58031
  • 23. D.SULLIVAN, Related aspects of positivity in Riemannian geometry, J. Differential Geom. 25 (1987), 327-351. MR 88d:58132
  • 24. W. P.THURSTON, The Geometry and Topology of 3-Manifolds, Princeton Lecture Notes, 1978.
  • 25. R.ZIMMER, Ergodic Theory and Semisimple Groups, Monographs in Mathematics, Vol. 81, Birkhäuser, Boston, 1984. MR 86j:22014

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Additional Information

Enrico Leuzinger
Affiliation: Math. Institut II, Universität Karlsruhe, D-76128 Karlsruhe, Germany
Email: Enrico.Leuzinger@math.uni-karlsruhe.de

DOI: https://doi.org/10.1090/S0002-9939-03-07173-9
Keywords: Discrete subgroups of semisimple Lie groups, critical exponent, $L^2$--spectrum, locally symmetric spaces
Received by editor(s): November 9, 2002
Published electronically: September 12, 2003
Communicated by: Rebecca A. Herb
Article copyright: © Copyright 2003 American Mathematical Society

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