manuscripta mathematica

, Volume 126, Issue 3, pp 375–391 | Cite as

Growth of conjugacy classes of Schottky groups in higher rank symmetric spaces

  • Gabriele LinkEmail author


Let X be a globally symmetric space of noncompact type and rank greater that one, and \({\Gamma \subset Isom(X)}\) a Schottky group of axial isometries. Then \({M := X/\Gamma}\) is a locally symmetric Riemannian manifold of infinite volume. The goal of this note is to give an asymptotic estimate for the number of primitive closed geodesics in M modulo free homotopy with period less than t.

Mathematics Subject Classification (2000)

20E45 22E40 53C35 


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  1. 1.
    Ballmann, W.: Lectures on spaces of nonpositive curvature, DMV Seminar, vol. 25. Birkhäuser Verlag, Basel, With an appendix by Misha Brin (1995)Google Scholar
  2. 2.
    Benoist Y.: Propriétés asymptotiques des groupes linéaires. Geom. Funct. Anal. 7(1), 1–47 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Ballmann W., Gromov M., Schroeder V.: Manifolds of nonpositive curvature, Progress in Mathematics, vol. 61. Birkhäuser Boston Inc, Boston (1985)Google Scholar
  4. 4.
    Coornaert M., Knieper G.: An upper bound for the growth of conjugacy classes in torsion-free word hyperbolic groups. Int. J. Algebra Comput. 14(4), 395–401 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    DeGeorge D.L.: Length spectrum for compact locally symmetric spaces of strictly negative curvature. Ann. Sci. École Norm. Sup. (4) 10(2), 133–152 (1977)zbMATHMathSciNetGoogle Scholar
  6. 6.
    de la Harpe P.: Free groups in linear groups. Enseign. Math. (2). 29(1–2), 129–144 (1983)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Dal’bo F., Peigné M.: Some negatively curved manifolds with cusps, mixing and counting. J. Reine Angew. Math. 497, 141–169 (1998)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Eberlein P.B.: Geometry of nonpositively curved manifolds. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1996)Google Scholar
  9. 9.
    Eskin A., McMullen C.: Mixing, counting, and equidistribution in Lie groups. Duke Math. J. 71(1), 181–209 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Helgason, S.: Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, vol. 34, American Mathematical Society, Providence, RI, Corrected reprint of the 1978 original (2001)Google Scholar
  11. 11.
    Knieper G.: Das Wachstum der Äquivalenzklassen geschlossener Geodätischer in kompakten Mannigfaltigkeiten. Arch. Math. Basel 40(6), 559–568 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Knieper G.: On the asymptotic geometry of nonpositively curved manifolds. Geom. Funct. Anal. 7(4), 755–782 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Knieper, G.: Hyperbolic dynamics and Riemannian geometry. Handbook of dynamical systems, vol. 1A, pp. 453–545. North-Holland, Amsterdam (2002)Google Scholar
  14. 14.
    Lalley S.P.: Renewal theorems in symbolic dynamics, with applications to geodesic flows, non-Euclidean tessellations and their fractal limits. Acta Math. 163(1–2), 1–55 (1989)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Link G.: Geometry and dynamics of discrete isometry groups of higher rank symmetric spaces. Geom. Dedicata 122, 51–75 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Link G.: Asymptotic geometry and growth of conjugacy classes of nonpositively curved manifolds. Ann. Global Anal. Geom. 31(1), 37–57 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Margulis G.A.: Applications of ergodic theory to the investigation of manifolds of negative curvature. Funkt. Anal. Appl. 3, 335–336 (1969)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Margulis G.A.: On some aspects of the theory of Anosov systems, Springer Monographs in Mathematics, Springer, Berlin, With a survey by Richard Sharp: Periodic orbits of hyperbolic flows, Translated from the Russian by Valentina Vladimirovna Szulikowska (2004)Google Scholar
  19. 19.
    Pollicott M., Sharp R.: Orbit counting for some discrete groups acting on simply connected manifolds with negative curvature. Invent. Math. 117(2), 275–302 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Quint J.-F.: Groupes de Schottky et comptage. Ann. Inst. Fourier (Grenoble) 55(2), 373–429 (2005)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Roblin T.: Sur la fonction orbitale des groupes discrets en courbure négative. Ann. Inst. Fourier (Grenoble). 52(1), 145–151 (2002)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Roblin, T.: Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr. (N.S.), no. 95, vi+96 (2003)Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Forschungsinstitut für MathematikZürichSwitzerland

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