Skip to main content
SpringerLink
Log in

Critical exponents of invariant random subgroups in negative curvature

  • Published:
Geometric and Functional Analysis Aims and scope Submit manuscript

Abstract

Let X be a proper geodesic Gromov hyperbolic metric space and let G be a cocompact group of isometries of X admitting a uniform lattice. Let d be the Hausdorff dimension of the Gromov boundary \({\partial X}\). We define the critical exponent \({\delta(\mu)}\) of any discrete invariant random subgroup \({\mu}\) of the locally compact group G and show that \({\delta(\mu) > \frac{d}{2}}\) in general and that \({\delta(\mu) = d}\) if \({\mu}\) is of divergence type. Whenever G is a rank-one simple Lie group with Kazhdan’s property (T) it follows that an ergodic invariant random subgroup of divergence type is a lattice. One of our main tools is a maximal ergodic theorem for actions of hyperbolic groups due to Bowen and Nevo.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abert, M., Bergeron, N., Biringer, I., Gelander, T., Nikolov, N., Raimbault, J., Samet, I.: On the growth of \(L^2\)-invariants for sequences of lattices in Lie groups. Annals of math 185(3), 711–790 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  2. G. Arzhantseva and C. Cashen. Cogrowth for group actions with strongly contracting elements. arXiv preprint arXiv:1803.05782, (2018)

  3. Abért, M., Glasner, Y., Virág, B.: Kesten's theorem for invariant random subgroups. Duke Mathematical Journal 163(3), 465–488 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bishop, C., Jones, P.: Hausdorff dimension and Kleinian groups. Acta Mathematica 179(1), 1–39 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bonk, M., Kleiner, B.: Rigidity for quasi-Moebius actions. Journal of Differential Geometry 61(1), 81–106 (2002)

    Article  MathSciNet  Google Scholar 

  6. L. Bowen and A. Nevo. von-Neumann and Birkhoff ergodic theorems for negatively curved groups. arXiv preprint arXiv:1303.4109, (2013)

  7. J. Cannizzo. Schreier Graphs and Ergodic Properties of Boundary Actions. PhD thesis, Université d'Ottawa/University of Ottawa (2014)

  8. R. Coulon, F. Dal'bo, and A. Sambusetti. Growth gap in hyperbolic groups and amenability. arXiv preprint arXiv:1709.07287, (2017)

  9. Coornaert, M.: Mesures de Patterson-Sullivan sur le bord dún espace hyperbolique au sens de Gromov. Pacific Journal of Mathematics 159(2), 241–270 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  10. Corlette, K.: Hausdorff dimensions of limit sets I. Inventiones mathematicae 102(1), 521–541 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  11. T. Das, D. Simmons, and M. Urbański. Geometry and Dynamics in Gromov Hyperbolic Metric Spaces, volume 218. American Mathematical Soc., (2017)

  12. J. Elstrodt. Die resolvente zum eigenwertproblem der automorphen formen in der hyperbolischen ebene. teil i. Mathematische Annalen, 203(4) (1973), 295–330

  13. J. Elstrodt. Die resolvente zum eigenwertproblem der automorphen formen in der hyperbolischen ebene. teil ii. Mathematische Zeitschrift, 132(2) (1973), 99–134

  14. J. Elstrodt. Die resolvente zum eigenwertproblem der automorphen formen in der hyperbolischen ebene. teil iii. Mathematische Annalen, 208(2) (1974), 99–132

  15. Grigorchuk, R., Kaimanovich, V., Nagnibeda, T.: Ergodic properties of boundary actions and the Nielsen-Schreier theory. Advances in Mathematics 230(3), 1340–1380 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. T. Gelander and A. Levit. Invariant random subgroups over non-Archimedean local fields. arXiv preprint arXiv:1707.03578, (2017)

  17. Hartman, Y., Tamuz, O.: Stabilizer rigidity in irreducible group actions. Israel Journal of Mathematics 216(2), 679–705 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jaerisch, J.: A lower bound for the exponent of convergence of normal subgroups of Kleinian groups. The Journal of Geometric Analysis 25(1), 298–305 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kechris, A.: Classical descriptive set theory, vol. 156. Springer, Berlin (2012)

    MATH  Google Scholar 

  20. Kesten, H.: Symmetric random walks on groups. Transactions of the American Mathematical Society 92(2), 336–354 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  21. E. Leuzinger. Critical exponents of discrete groups and \(l^2\)–spectrum. Proceedings of the American Mathematical Society, (3)132 (2004), 919–927

  22. K. Matsuzaki. Isoperimetric constants for conservative Fuchsian groups. Kodai Mathematical Journal, (2)28 (2005), 292–300

  23. K. Matsuzaki, Y. Yabuki, and J. Jaerisch. Normalizer, divergence type and Patterson measure for discrete groups of the Gromov hyperbolic space. arXiv preprint arXiv:1511.02664, (2015)

  24. K. Ōshika. Discrete groups, volume 207. American Mathematical Soc., 2002

  25. S. Patterson. The limit set of a Fuchsian group. Acta mathematica, (1)136 (1976), 241–273

  26. S. Patterson. Spectral theory and Fuchsian groups. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 81, pages 59–75. Cambridge University Press, (1977)

  27. Patterson, S.: Further remarks on the exponent of convergence of Poincaré series. Tohoku Math Journal 35(2), 357–373 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  28. Quint, J.-F.: An overview of Patterson-Sullivan theory. In Workshop The barycenter method, FIM, Zurich (2006)

    Google Scholar 

  29. T. Roblin. Ergodicité et équidistribution en courbure négative. Société mathématique de France, (2003)

  30. Sullivan, D.: The density at infinity of a discrete group of hyperbolic motions. Publications Mathématiques de l'Institut des Hautes Études Scientifiques 50(1), 171–202 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  31. Sullivan, D.: Related aspects of positivity in Riemannian geometry. Journal of differential geometry 25(3), 327–351 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  32. Stuck, G., Zimmer, R.J.: Stabilizers for ergodic actions of higher rank semisimple groups. Annals of Mathematics 723–747 (1994)

  33. Zimmer, R.J.: Ergodic theory and semisimple groups, vol. 81. Springer, Berlin (2013)

    MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank Mikolaj Fraczyk for pointing out the analogy with Kesten’s theorem and leading us to include Corollary 1.3. We would like to thank Amos Nevo for useful remarks and suggestions concerning the ergodic theorem for hyperbolic groups. We would like to thank Tushar Das and David Simmons for illuminating remarks about critical exponents, especially alerting us to Patterson’s construction of Kleinian groups with full limit set and arbitrarily small critical exponent. The first author is partially supported by NSF Grant DMS-1401875.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Toronto, 40 St. George St, Toronto, ON, Canada

    Ilya Gekhtman

  2. Department of Mathematics, Yale University, 12 Hillhouse Avenue, New Haven, CT, USA

    Arie Levit

Authors
  1. Ilya Gekhtman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Arie Levit
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ilya Gekhtman.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gekhtman, I., Levit, A. Critical exponents of invariant random subgroups in negative curvature. Geom. Funct. Anal. 29, 411–439 (2019). https://doi.org/10.1007/s00039-019-00485-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00039-019-00485-5

Search

Navigation