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The Martin boundary of an extension by a hyperbolic group

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Abstract

We prove uniform Ancona—Gouëzel—Lalley inequalities for an extension by a hyperbolic group G of a Markov map which allows to deduce that the visual boundary of the group and the Martin boundary are Hölder equivalent. As application, we identify the set of minimal δ-conformal measures of a regular cover of a convex-cocompact CAT(-1)-manifold with the visual boundary of the covering group, provided that this group is hyperbolic. In this setting, the uniformity allows to identify the visual boundary with minimal, δ-conformal densities in the sense of Patterson with respect to the exponent of convergence δ. This is of interest as δ coincides with the Hausdorff dimension of the radial limit set. Moreover, this extends Roblin’s identification of s-conformal densities for s > δ.

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References

  1. J. Aaronson, An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs, Vol. 50, American Mathematical Society, Providence, RI, 1997.

    MATH  Google Scholar 

  2. J. Aaronson, M. Denker and M. Urbański, Ergodic theory for Markov fibred systems and parabolic rational maps, Transactions of the American Mathematical Society 337 (1993), 495–548.

    Article  MathSciNet  MATH  Google Scholar 

  3. T. Adachi, Markov families for anosov flows with an involutive action, Nagoya Mathematical Journal 104 (1986), 55–62.

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Ancona, Negatively curved manifolds, elliptic operators, and the Martin boundary, Annals of Mathematics 125 (1987), 495–536.

    Article  MathSciNet  MATH  Google Scholar 

  5. J. W. Anderson, K. Falk and P. Tukia, Conformal measures associated to ends of hyperbolic n-manifolds, Quarterly Journal of Mathematics 58 (2007), 1–15.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. R. Brooks, The bottom of the spectrum of a Riemannian covering, Journal für die Reine und Angewandte Mathematik 357 (1985), 101–114.

    MathSciNet  MATH  Google Scholar 

  8. P.-E. Caprace, Y. Cornulier, N. Monod and R. Tessera, Amenable hyperbolic groups, Journal of the European Mathematical Society 17 (2015), 2903–2947.

    Article  MathSciNet  MATH  Google Scholar 

  9. D. Constantine, J.-F. Lafont and D. J. Thompson, Strong symbolic dynamics for geodesic flow on CAT(−1) spaces and other metric Anosov flows, Journal de l’École polytechnique. Mathématiques 7 (2020), 201–231.

    Article  MathSciNet  MATH  Google Scholar 

  10. M. Coornaert and A. Papadopoulos, Horofunctions and symbolic dynamics on Gromov hyperbolic groups, Glasgow Mathematical Journal 43 (2001), 425–456.

    Article  MathSciNet  MATH  Google Scholar 

  11. R. Coulon, F. Dal’Bo and A. Sambusetti, Growth gap in hyperbolic groups and amenability, Geometric and Functional Analysis 28 (2018), 1260–1320.

    Article  MathSciNet  MATH  Google Scholar 

  12. T. Das, D. Simmons and M. Urbański, Geometry and Dynamics in Gromov Hyperbolic Metric Spaces: With an Emphasis on Non-Proper Settings, Mathematical Surveys and Monographs, Vol. 218, American Mathematical Society, Providence, RI, 2017.

    MATH  Google Scholar 

  13. R. Dougall and R. Sharp, Amenability, critical exponents of subgroups and growth of closed geodesics, Mathematische Annalen 365 (2016), 1359–1377.

    Article  MathSciNet  MATH  Google Scholar 

  14. K. Falk, K. Matsuzaki and B. O. Stratmann, Checking atomicity of conformal ending measures for kleinian groups, Conformal Geometry and Dynamics 14 (2010), 167–183.

    Article  MathSciNet  MATH  Google Scholar 

  15. E. Ghys and P. de la Harpe, Espaces métriques hyperboliques, in Sur les Groupes Hyperboliques d’après Mikhael Gromov, Progress in Mathematics, Vol. 83, Birkhäuser, Basel, 1990, pp. 27–46.

    Chapter  MATH  Google Scholar 

  16. E. Ghys and P. de la Harpe, Le bord d’un espace hyperbolique, in Sur les Groupes Hyperboliques d’après Mikhael Gromov, Progress in Mathematics, Vol. 83, Birkhäuser, Basel, 1990, pp. 117–134.

    Chapter  MATH  Google Scholar 

  17. S. Gouëezel, Local limit theorem for symmetric random walks in Gromov-hyperbolic groups, Journal of the American Mathematical Society 27 (2014), 893–928.

    Article  MathSciNet  Google Scholar 

  18. S. Gouëzel, Martin boundary of random walks with unbounded jumps in hyperbolic groups, Annals of Probability 43 (2015), 2374–2404.

    Article  MathSciNet  MATH  Google Scholar 

  19. S. Gouëzel and S. P. Lalley, Random walks on co-compact Fuchsian groups, Annales Scientifiques de l’École Normale Supérieure 46 (2013), 129–173.

    MathSciNet  MATH  Google Scholar 

  20. M. Gromov, Hyperbolic groups, in Essays in Group Theory, Mathematical Sciences Research Institute Publications, Vol. 8, Springer, New York, 1987, pp. 75–263.

    Chapter  Google Scholar 

  21. J. Jaerisch, Recurrence and pressure for group extensions, Ergodic Theory and Dynamical Systems 26 (2016), 108–126.

    Article  MathSciNet  MATH  Google Scholar 

  22. V. A. Kaimanovich, Ergodic properties of the horocycle flow and classification of Fuchsian groups, Journal of Dynamical and Control Systems 6 (2000), 21–56.

    Article  MathSciNet  MATH  Google Scholar 

  23. A. Karlsson and G. A. Margulis, A multiplicative ergodic theorem and nonpositively curved spaces, Communications in Mathematical Physics 208 (1999), 107–123.

    Article  MathSciNet  MATH  Google Scholar 

  24. H. Kesten, Full Banach mean values on countable groups, Mathematica Scandinavica 7 (1959), 146–156.

    Article  MathSciNet  MATH  Google Scholar 

  25. R. S. Martin, Minimal positive harmonic functions, Transactions of the American Mathematical Society 49 (1941), 137–172.

    Article  MathSciNet  MATH  Google Scholar 

  26. M. Murata, Martin boundaries of elliptic skew products, semismall perturbations, and fundamental solutions of parabolic equations, Journal of Functional Analysis 194 (2002), 53–141.

    Article  MathSciNet  MATH  Google Scholar 

  27. M. Rees, Checking ergodicity of some geodesic flows with infinite Gibbs measure, Ergodic Theory and Dynamical Systems 1 (1981), 107–133.

    Article  MathSciNet  MATH  Google Scholar 

  28. D. Revuz, Markov chains, North-Holland Mathematical Library, Vol. 11, North-Holland, Amsterdam, 1984.

    MATH  Google Scholar 

  29. T. Roblin, Ergodicité et équidistribution en courbure négative, Mémoires de la Société Mathematique de France 95 (2003).

  30. T. Roblin, Comportement harmonique des densités conformes et frontière de Martin, Bulletin de la Société Mathématique de France 139 (2011), 97–128.

    Article  MathSciNet  MATH  Google Scholar 

  31. O. M. Sarig, Existence of Gibbs measures for countable Markov shifts, Proceedings of the American Mathematical Society 131 (2003), 1751–1758.

    Article  MathSciNet  MATH  Google Scholar 

  32. O. Shwartz, Thermodynamic formalism for transient potential functions, Communications in Mathematical Physics 366 (2019), 737–779.

    Article  MathSciNet  MATH  Google Scholar 

  33. O. Shwartz, The conformal measures of a normal subgroup of a cocompact Fuchsian group, Ergodic Theory and Dynamical Systems 41 (2021), 2845–2878.

    Article  MathSciNet  MATH  Google Scholar 

  34. M. Stadlbauer, An extension of Kesten’s criterion for amenability to topological Markov chains, Advances in Mathematics 235 (2013), 450–468.

    Article  MathSciNet  MATH  Google Scholar 

  35. M. Stadlbauer, On conformal measures and harmonic functions for group extensions, in New Trends in One-Dimensional Dynamics, Springer Proceedings in Mathematics & Statistics, Vol. 285, Springer, Cham, 2019, pp. 271–299.

    Chapter  Google Scholar 

  36. W. Woess, Random walks on infinite graphs and groups—a survey on selected topics, Bulletin of the London Mathematical Society 26 (1994), 1–60.

    Article  MathSciNet  MATH  Google Scholar 

  37. W. Woess, Denumerable Markov chains, EMS Textbooks in Mathematics, European Mathematical Society, Zürich, 2009.

    Book  MATH  Google Scholar 

  38. R. J. Zimmer, Amenable ergodic group actions and an application to Poisson boundaries of random walks, Journal of Functional Analysis 27 (1978), 350–372.

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to express their deep gratitude for support of the Post-Graduate program of the Universidade Federal da Bahia, where the first author obtained her PhD in 2019 under the supervision of the second author. In particular, the authors acknowledge support by CAPES and CNPq: The first author was supported by CAPES during her PhD studies, and the second author was partially supported by CAPES (Programa PROEX da Pós-Graduação em Matemática do IM-UFRJ) and CNPq (PQ 312632/2018-5, Universal 426814/2016-9).

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Authors and Affiliations

  1. Centro das Ciências Exatas e das Tecnologias, Universidade Federal do Oeste da Bahia, 47808-021, Barreiras (BA), Brazil

    Sara Ruth Pires Bispo

  2. Departamento de Matemática, Universidade Federal do Rio de Janeiro, 21941-909, Rio de Janeiro (RJ), Brazil

    Manuel Stadlbauer

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  1. Sara Ruth Pires Bispo
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  2. Manuel Stadlbauer
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Correspondence to Manuel Stadlbauer.

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Bispo, S.R.P., Stadlbauer, M. The Martin boundary of an extension by a hyperbolic group. Isr. J. Math. 255, 1–62 (2023). https://doi.org/10.1007/s11856-023-2468-x

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  • DOI: https://doi.org/10.1007/s11856-023-2468-x

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