Skip to main content
SpringerLink
Log in

Stability phenomena for Martin boundaries of relatively hyperbolic groups

  • Published:
Probability Theory and Related Fields Aims and scope Submit manuscript

Abstract

Let \(\Gamma \) be a relatively hyperbolic group and let \(\mu \) be an admissible symmetric finitely supported probability measure on \(\Gamma \). We extend Floyd–Ancona type inequalities from Gekhtman et al. (Martin boundary covers Floyd boundary, 2017. arXiv:1708.02133) up to the spectral radius R of \(\mu \). We use them to find the precise homeomorphism type of the r-Martin boundary, which describes r-harmonic functions, for every \(r\le R\). We also define a notion of spectral degeneracy along parabolic subgroups which is crucial to describe the homeomorphism type of the R-Martin boundary. Finally, we give a criterion for (strong) stability of the Martin boundary in the sense of Picardello and Woess (in: Potential theory, de Gruyter, 1992) in terms of spectral degeneracy. We then prove that this criterion is always satisfied in small rank, so that in particular, the Martin boundary of an admissible symmetric finitely supported probability measure on a geometrically finite Kleinian group of dimension at most 5 is always strongly stable.

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. Ancona, A.: Positive harmonic functions and hyperbolicity. In: Potential Theory-Surveys and Problems. Lecture Notes in Mathematics, pp. 1–23. Springer (1988)

  2. Ancona, A.: Théorie du potentiel sur les graphes et les variétés. In: École d’été de Probabilités de Saint-Flour XVIII. Lecture Notes in Mathematics, pp. 1–112. Springer (1990)

  3. Bowditch, B.: Geometrical finiteness for hyperbolic groups. J. Funct. Anal. 113, 245–317 (1993)

    Article  MathSciNet  Google Scholar 

  4. Bowditch, B.: Relatively hyperbolic group. Int. J. Algebra Comput. 22, 66 (2012)

    Article  MathSciNet  Google Scholar 

  5. Bridson, M., Häfliger, A.: Metric Spaces of Non-positive Curvature. Springer, New York (1999)

    Book  Google Scholar 

  6. Candellero, E., Gilch, L.A.: Phase transitions for random walk asymptotics on free products of groups. Random Struct. Algorithms 40, 150–181 (2012)

    Article  MathSciNet  Google Scholar 

  7. Dahmani, F.: Classifying spaces and boundaries for relatively hyperbolic groups. Proc. Lond. Math. Soc. 86, 666–684 (2003)

    Article  MathSciNet  Google Scholar 

  8. Dussaule, M.: The Martin boundary of a free product of Abelian groups (2017). arXiv:1709.07738

  9. Dussaule, M.: Local limit theorems in relatively hyperbolic groups II: the non spectrally degenerate case (2020). arXiv:2004.13986

  10. Dussaule, M., Gekhtman, I.: Entropy and drift for word metric on relatively hyperbolic groups (2018). arXiv:1811.10849

  11. Dussaule, M., Gekhtman, I., Gerasimov, V., Potyagailo, L.: The Martin boundary of relatively hyperbolic groups with virtually Abelian parabolic subgroups (2017). arXiv:1711.11307

  12. Gekhtman, I., Gerasimov, V., Potyagailo, L., Yang, W.: Martin boundary covers Floyd boundary (2017). arXiv:1708.02133

  13. Gerasimov, V.: Expansive convergence groups are relatively hyperbolic. Geom. Funct. Anal. 19, 137–169 (2009)

    Article  MathSciNet  Google Scholar 

  14. Gerasimov, V.: Floyd maps for relatively hyperbolic groups. Geom. Funct. Anal. 22, 1361–1399 (2012)

    Article  MathSciNet  Google Scholar 

  15. Gerasimov, V., Potyagailo, L.: Quasi-isometries and Floyd boundaries of relatively hyperbolic groups. J. Eur. Math. Soc. 15, 2115–2137 (2013)

    Article  Google Scholar 

  16. Gerasimov, V., Potyagailo, L.: Quasiconvexity in the relatively hyperbolic groups. J. Pure Appl. Math. 710, 95–135 (2016)

    MathSciNet  MATH  Google Scholar 

  17. Gouëzel, S.: Local limit theorem for symmetric random walks in Gromov-hyperbolic groups. J. Am. Math. Soc. 27, 893–928 (2014)

    Article  MathSciNet  Google Scholar 

  18. Gouëzel, S., Lalley, S.: Random walks on co-compact Fuchsian groups. Annales Scientifiques de l’ENS 46, 129–173 (2013)

    MathSciNet  MATH  Google Scholar 

  19. Gouëzel, S., Mathéus, F., Maucourant, F.: Entropy and drift in word hyperbolic groups. Invent. Math. 211, 1201–1255 (2018)

    Article  MathSciNet  Google Scholar 

  20. Guivarc’h, Y.: Sur la loi des grands nombres et le rayon spectral d’une marche aléatoire. In: Conference on Random Walks. Astérisque, vol. 74, pp. 47–98. Soc. Math. France (1980)

  21. Hruska, G.: Relative hyperbolicity and relative quasiconvexity for countable groups. Algebraic Geom. Topol. 10, 1807–1856 (2010)

    Article  MathSciNet  Google Scholar 

  22. Izumi, M., Neshveyev, S., Okayasu, R.: The ratio set of the harmonic measure of a random walk on a hyperbolic group. Israel J. Math. 163, 285–316 (2008)

    Article  MathSciNet  Google Scholar 

  23. Kaimanovich, V.: Boundaries of invariant Markov operators: the identification problem. In: Ergodi Theory of \(\mathbb{Z}^d\) Actions. London Mathematical Society Lecture Note Series, vol. 228, pp. 127–176. Cambridge University Press (1996)

  24. Kaimanovich, V.: The Poisson formula for groups with hyperbolic properties. Ann. Math. 152, 659–692 (2000)

    Article  MathSciNet  Google Scholar 

  25. Kaimanovich, V., Vershik, A.: Random walks on discrete groups: boundary and entropy. Ann. Probab. 11, 457–490 (1983)

    Article  MathSciNet  Google Scholar 

  26. Karlsson, A.: Free subgroups of groups with nontrivial Floyd boundary. Commun. Algebra 31, 5361–5376 (2003)

    Article  MathSciNet  Google Scholar 

  27. Ledrappier, F.: Regularity of the entropy for random walks on hyperbolic groups. Ann. Probab. 41, 3582–3605 (2013)

    Article  MathSciNet  Google Scholar 

  28. Margulis, G.: Positive harmonic functions on nilpotent groups. Soviet Math. Doklady 7, 241–244 (1966)

    MathSciNet  MATH  Google Scholar 

  29. Ney, P., Spitzer, F.: The Martin boundary for random walk. Trans. Am. Math. Soc. 11, 116–132 (1966)

    Article  MathSciNet  Google Scholar 

  30. Picardello, M., Woess, W.: Examples of stable Martin boundaries of Markov chains. In: Potential theory, pp. 261–270. de Gruyter (1992)

  31. Ratcliffe, J.: Foundations of Hyperbolic Manifolds. Graduate Texts in Mathematics. Springer, New York (2006)

    MATH  Google Scholar 

  32. Sawyer, S.: Martin boundaries and random walks. Contemp. Math. 206, 17–44 (1997)

    Article  MathSciNet  Google Scholar 

  33. Seneta, E.: Non-negative Matrices and Markov Chains. Springer, New York (1981)

    Book  Google Scholar 

  34. Series, C.: Martin boundaries of random walks on Fuchsian groups. Israel J. Math. 44, 221–242 (1983)

    Article  MathSciNet  Google Scholar 

  35. Sisto, A.: Projections and relative hyperbolicity. L’Enseignement Mathématique 59, 165–181 (2013)

    Article  MathSciNet  Google Scholar 

  36. Vershik, A.: Dynamic theory of growth in groups: entropy, boundaries, examples. Uspekhi Matematicheskikh Nauk 55, 59–128 (2000)

    Article  MathSciNet  Google Scholar 

  37. Woess, W.: A description of the Martin boundary for nearest neighbour random walks on free products. In: Probability Measures on Groups VIII. Lecture Notes in Mathematics, pp. 203–215. Springer (1986)

  38. Woess, W.: Boundaries of random walks on graphs and groups with infinitely many ends. Israel J. Math. 68, 271–301 (1989)

    Article  MathSciNet  Google Scholar 

  39. Woess, W.: Random Walks on Infinite Graphs and Groups. Cambridge Press University, Cambridge (2000)

    Book  Google Scholar 

  40. Yaman, A.: A topological characterisation of relatively hyperbolic groups. J. Pure Appl. Math. 566, 41–89 (2004)

    MathSciNet  MATH  Google Scholar 

  41. Yang, W.: Patterson-Sullivan measures and growth of relatively hyperbolic groups (2013). arXiv:1308.6326

  42. Yang, W.: Growth tightness for groups with contracting elements. Math. Proc. Camb. Philos. Soc. 157, 297–319 (2014)

    Article  MathSciNet  Google Scholar 

  43. Yang, W.: Statistically convex-cocompact actions of groups with contracting elements. Int. Math. Res. Not. 00, 1–65 (2018)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universite de Nantes, Nantes, France

    Matthieu Dussaule

  2. Technion, Haifa, Israel

    Ilya Gekhtman

Authors
  1. Matthieu Dussaule
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ilya Gekhtman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matthieu Dussaule.

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

Dussaule, M., Gekhtman, I. Stability phenomena for Martin boundaries of relatively hyperbolic groups. Probab. Theory Relat. Fields 179, 201–259 (2021). https://doi.org/10.1007/s00440-020-01000-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00440-020-01000-w

Mathematics Subject Classification

Search

Navigation