Mathematische Zeitschrift

, Volume 279, Issue 3–4, pp 1175–1196 | Cite as

On the hyperbolic orbital counting problem in conjugacy classes



Given a discrete group \(\Gamma \) of isometries of a negatively curved manifold \({\widetilde{M}}\), a non-trivial conjugacy class \({\mathfrak {K}}\) in \(\Gamma \) and \(x_0\in {\widetilde{M}}\), we give asymptotic counting results, as \(t\rightarrow +\infty \), on the number of orbit points \(\gamma x_0\) at distance at most \(t\) from \(x_0\), when \(\gamma \) is restricted to be in \({\mathfrak {K}}\), as well as related equidistribution results. These results generalise and extend work of Huber on cocompact hyperbolic lattices in dimension 2. We also study the growth of given conjugacy classes in finitely generated groups endowed with a word metric.


Counting Equidistribution Hyperbolic geometry Growth Conjugacy class 

Mathematics Subject Classification

37C35 20H10 30F40 53A35 20G20 


  1. 1.
    Athreya, J., Bufetov, A., Eskin, A., Mirzakhani, M.: Lattice point asymptotics and volume growth on Teichmüller space. Duke Math. J. 161, 1055–1111 (2012)Google Scholar
  2. 2.
    Babillot, M.: On the mixing property for hyperbolic systems. Israel J. Math. 129, 61–76 (2002)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Babillot, M.: Points entiers et groupes discrets: de l’analyse aux systèmes dynamiques. In: “Rigidité, groupe fondamental et dynamique”, Panor. Synthèses, vol. 13, pp. 1–119. Société Mathématique, France (2002)Google Scholar
  4. 4.
    Beardon, A.F.: The Geometry of Discrete Groups. Graduate Texts in Mathematics, vol. 91. Springer, New York (1983)CrossRefGoogle Scholar
  5. 5.
    Bourgain, J., Kontorovich, A., Sarnak, P.: Sector estimates for hyperbolic isometries. Geom. Funct. Anal. 20, 1175–1200 (2010)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Bridson, M.R., Haefliger, A.: Metric Spaces of Non-positive Curvature. Grund. Math. Wiss, vol. 319. Springer (1999)Google Scholar
  7. 7.
    Broise-Alamichel, A., Parkkonen, J., Paulin, F.: Counting paths in graphs. In preparationGoogle Scholar
  8. 8.
    Champetier, C.: L’espace des groupes de type fini. Topology 39, 657–680 (2000)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Chen, S.S., Greenberg, L.: Hyperbolic spaces. In: Contributions to Analysis (a collection of papers dedicated to Lipman Bers), pp. 49–87. Academic Press (1974)Google Scholar
  10. 10.
    Dal’Bo, F.: Remarques sur le spectre des longueurs d’une surface et comptage. Bol. Soc. Bras. Math. 30, 199–221 (1999)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Dal’Bo, F., Otal, J.-P., Peigné, M.: Séries de Poincaré des groupes géométriquement finis. Israel J. Math. 118, 109–124 (2000)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Douma, F.: A lattice point problem on the regular tree. Discrete Math. 311, 276–281 (2011)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Fenchel, W.: Elementary Geometry in Hyperbolic Space. de Gruyter Stud. Math, vol. 11. de Gruyter (1989)Google Scholar
  14. 14.
    Goldman, W.M.: Complex Hyperbolic Geometry. Oxford University Press (1999)Google Scholar
  15. 15.
    Grigorchuk, R.: Milnor’s problem on the growth of groups and its consequences. In: Bonifant, A., Lyubich, M., Sutherland, S. (eds.) Frontiers in Complex Dynamics. In Celebration of John Milnor’s 80th Birthday. Princeton University Press (2014)Google Scholar
  16. 16.
    Guba, V., Sapir, M.: On the conjugacy growth functions of groups. Illinois J. Math. 54, 301–313 (2010)MATHMathSciNetGoogle Scholar
  17. 17.
    de la Harpe, P.: Topics in Geometric Group Theory. Chicago University Press (2000)Google Scholar
  18. 18.
    Huber, H.: Über eine neue Klasse automorpher Funktionen und ein Gitterpunktproblem in der hyperbolischen Ebene I. Comment. Math. Helv. 30, 20–62 (1956)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Huber, H.: Ein Gitterpunktproblem in der hyperbolischen Ebene. J. reine angew. Math. 496, 15–53 (1998)MATHMathSciNetGoogle Scholar
  20. 20.
    Hull, M., Osin, D.: Conjugacy growth of finitely generated groups. Adv. Math. 235, 361–389 (2013)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Kenison, G., Sharp. R.: Orbit counting in conjugacy classes for free groups acting on trees. (preprint, 2015)Google Scholar
  22. 22.
    Mann, A.: How Groups Grow. London Math. Soc. Lect. Note Ser., vol. 395. Cambridge University Press (2012)Google Scholar
  23. 23.
    Mohammadi, A., Oh, H.: Matrix coefficients, counting and primes for orbits of geometrically finite groups. Preprint arXiv:1208.4139. To appear in Journal of European Mathematical Society
  24. 24.
    Oh, H.: Orbital counting via mixing and unipotent flows. In: Einsiedler, M., et al. (eds.) Homogeneous Flows, Moduli Spaces and Arithmetic. Clay Math. Proc., vol. 10, pp. 339–375. American Mathematical Society (2010)Google Scholar
  25. 25.
    Oh, H.: Harmonic analysis, ergodic theory and counting for thin groups. In: Breuillard, E., Oh, H. (eds.) Thin groups and Superstrong Approximation, M.S.R.I. Publ., vol. 61. Cambridge University Press (2014)Google Scholar
  26. 26.
    Oh, H., Shah, N.: Equidistribution and counting for orbits of geometrically finite hyperbolic groups. J. Am. Math. Soc. 26, 511–562 (2013)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Oh, H., Shah, N.: Counting visible circles on the sphere and Kleinian groups. Preprint arXiv:1004.2129. To appear in Aravinda, C.S., Farrell, T., Lafont, J.-F. (eds.) Geometry, Topology and Dynamics in Negative Curvature (ICM 2010 satellite conference, Bangalore), London Math. Soc, Lect. Notes
  28. 28.
    Ol’shanskii, A.: Geometry of Defining Relations in Groups. Mathematics and its Applications (Soviet Series), vol. 70. Kluwer (1991)Google Scholar
  29. 29.
    Parkkonen, J., Paulin, F.: Prescribing the behaviour of geodesics in negative curvature. Geom. Topol. 14, 277–392 (2010)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Parkkonen, J., Paulin, F.: Skinning measure in negative curvature and equidistribution of equidistant submanifolds. Ergod. Theory Dyn. Syst. 34, 1310–1342 (2014)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Parkkonen, J., Paulin, F.: On the arithmetic of crossratios and generalised Mertens’ formulas. In Dal’Bo, F., Lecuire, C. (eds.) numéro spécial “Aux croisements de la géométrie hyperbolique et de l’arithmétique”, Ann. Fac. Sci. Toulouse 23, 967–1022 (2014)Google Scholar
  32. 32.
    Parkkonen, J., Paulin, F.: Counting arcs in negative curvature. Preprint arXiv:1203.0175. To appear in Aravinda, C.S., Farrell, T. Lafont, J.-F. (eds.) Geometry, Topology and Dynamics in Negative Curvature (ICM 2010 satellite conference, Bangalore), London Math. Soc. Lect. Notes
  33. 33.
    Parkkonen, J., Paulin, F.: Counting common perpendicular arcs in negative curvature. Preprint arXiv:1305.1332
  34. 34.
    Parkkonen, J., Paulin, F.: Counting and equidistribution in the Heisenberg group. Preprint hal-00955576, arXiv:1402.7225
  35. 35.
    Parkkonen, J., Paulin, F.: A survey of some arithmetic applications of ergodic theory in negative curvature. Preprint hal-01102065v1, arXiv:1501.02072
  36. 36.
    Paulin, F., Pollicott, M., Schapira, B.: Equilibrium states in negative curvature. Book preprint arXiv:1211.6242. To appear in Astérisque, Soc. Math. France
  37. 37.
    Petridis, Y.N., Risager, M.S.: Hyperbolic lattice-point counting and modular symbols. J. Theo. Nomb. Bordeaux 21, 719–732 (2009)MATHMathSciNetGoogle Scholar
  38. 38.
    Quint, J.-F.: Groupes de Schottky et comptage. Ann. Inst. Fourier 55, 373–429 (2005)CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    Roblin, T.: Sur la fonction orbitale des groupes discrets en courbure négative. Ann. Inst. Fourier 52, 145–151 (2002)CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    Roblin, T.: Ergodicité et équidistribution en courbure négative, Mémoire, vol. 95. Soc. Math. France (2003)Google Scholar
  41. 41.
    Sambarino, A.: The orbital counting problem for hyperconvex representations. Preprint arXiv:1203.0280

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of JyväskyläUniversity of JyväskyläFinland
  2. 2.Département de mathématique, UMR 8628 CNRS, Bât. 425Université Paris-SudOrsay CedexFrance

Personalised recommendations