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.
Similar content being viewed by others
References
Ancona, A.: Positive harmonic functions and hyperbolicity. In: Potential Theory-Surveys and Problems. Lecture Notes in Mathematics, pp. 1–23. Springer (1988)
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)
Bowditch, B.: Geometrical finiteness for hyperbolic groups. J. Funct. Anal. 113, 245–317 (1993)
Bowditch, B.: Relatively hyperbolic group. Int. J. Algebra Comput. 22, 66 (2012)
Bridson, M., Häfliger, A.: Metric Spaces of Non-positive Curvature. Springer, New York (1999)
Candellero, E., Gilch, L.A.: Phase transitions for random walk asymptotics on free products of groups. Random Struct. Algorithms 40, 150–181 (2012)
Dahmani, F.: Classifying spaces and boundaries for relatively hyperbolic groups. Proc. Lond. Math. Soc. 86, 666–684 (2003)
Dussaule, M.: The Martin boundary of a free product of Abelian groups (2017). arXiv:1709.07738
Dussaule, M.: Local limit theorems in relatively hyperbolic groups II: the non spectrally degenerate case (2020). arXiv:2004.13986
Dussaule, M., Gekhtman, I.: Entropy and drift for word metric on relatively hyperbolic groups (2018). arXiv:1811.10849
Dussaule, M., Gekhtman, I., Gerasimov, V., Potyagailo, L.: The Martin boundary of relatively hyperbolic groups with virtually Abelian parabolic subgroups (2017). arXiv:1711.11307
Gekhtman, I., Gerasimov, V., Potyagailo, L., Yang, W.: Martin boundary covers Floyd boundary (2017). arXiv:1708.02133
Gerasimov, V.: Expansive convergence groups are relatively hyperbolic. Geom. Funct. Anal. 19, 137–169 (2009)
Gerasimov, V.: Floyd maps for relatively hyperbolic groups. Geom. Funct. Anal. 22, 1361–1399 (2012)
Gerasimov, V., Potyagailo, L.: Quasi-isometries and Floyd boundaries of relatively hyperbolic groups. J. Eur. Math. Soc. 15, 2115–2137 (2013)
Gerasimov, V., Potyagailo, L.: Quasiconvexity in the relatively hyperbolic groups. J. Pure Appl. Math. 710, 95–135 (2016)
Gouëzel, S.: Local limit theorem for symmetric random walks in Gromov-hyperbolic groups. J. Am. Math. Soc. 27, 893–928 (2014)
Gouëzel, S., Lalley, S.: Random walks on co-compact Fuchsian groups. Annales Scientifiques de l’ENS 46, 129–173 (2013)
Gouëzel, S., Mathéus, F., Maucourant, F.: Entropy and drift in word hyperbolic groups. Invent. Math. 211, 1201–1255 (2018)
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)
Hruska, G.: Relative hyperbolicity and relative quasiconvexity for countable groups. Algebraic Geom. Topol. 10, 1807–1856 (2010)
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)
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)
Kaimanovich, V.: The Poisson formula for groups with hyperbolic properties. Ann. Math. 152, 659–692 (2000)
Kaimanovich, V., Vershik, A.: Random walks on discrete groups: boundary and entropy. Ann. Probab. 11, 457–490 (1983)
Karlsson, A.: Free subgroups of groups with nontrivial Floyd boundary. Commun. Algebra 31, 5361–5376 (2003)
Ledrappier, F.: Regularity of the entropy for random walks on hyperbolic groups. Ann. Probab. 41, 3582–3605 (2013)
Margulis, G.: Positive harmonic functions on nilpotent groups. Soviet Math. Doklady 7, 241–244 (1966)
Ney, P., Spitzer, F.: The Martin boundary for random walk. Trans. Am. Math. Soc. 11, 116–132 (1966)
Picardello, M., Woess, W.: Examples of stable Martin boundaries of Markov chains. In: Potential theory, pp. 261–270. de Gruyter (1992)
Ratcliffe, J.: Foundations of Hyperbolic Manifolds. Graduate Texts in Mathematics. Springer, New York (2006)
Sawyer, S.: Martin boundaries and random walks. Contemp. Math. 206, 17–44 (1997)
Seneta, E.: Non-negative Matrices and Markov Chains. Springer, New York (1981)
Series, C.: Martin boundaries of random walks on Fuchsian groups. Israel J. Math. 44, 221–242 (1983)
Sisto, A.: Projections and relative hyperbolicity. L’Enseignement Mathématique 59, 165–181 (2013)
Vershik, A.: Dynamic theory of growth in groups: entropy, boundaries, examples. Uspekhi Matematicheskikh Nauk 55, 59–128 (2000)
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)
Woess, W.: Boundaries of random walks on graphs and groups with infinitely many ends. Israel J. Math. 68, 271–301 (1989)
Woess, W.: Random Walks on Infinite Graphs and Groups. Cambridge Press University, Cambridge (2000)
Yaman, A.: A topological characterisation of relatively hyperbolic groups. J. Pure Appl. Math. 566, 41–89 (2004)
Yang, W.: Patterson-Sullivan measures and growth of relatively hyperbolic groups (2013). arXiv:1308.6326
Yang, W.: Growth tightness for groups with contracting elements. Math. Proc. Camb. Philos. Soc. 157, 297–319 (2014)
Yang, W.: Statistically convex-cocompact actions of groups with contracting elements. Int. Math. Res. Not. 00, 1–65 (2018)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-020-01000-w