Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Ahlfors regularity of Patterson-Sullivan measures of Anosov groups and applications (2401.12398v4)

Published 22 Jan 2024 in math.GR, math.DS, math.GT, math.MG, and math.SP

Abstract: For all Zarski dense Anosov subgroups of a semisimple real algebraic group, we prove that their limit sets are Ahlfors regular for intrinsic conformal premetrics. As a consequence, we obtain that a Patterson-Sullivan measure is equal to the Hausdorff measure if and only if the associated linear form is symmetric. We also discuss several applications, including analyticity of $(p,q)$-Hausdorff dimensions on the Teichm\"uller spaces, new upper bounds on the growth indicator, and $L2$-spectral properties of associated locally symmetric manifolds.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (51)
  1. P. Albuquerque. Patterson-Sullivan theory in higher rank symmetric spaces. Geom. Funct. Anal., 9(1):1–28, 1999.
  2. Y. Benoist. Propriétés asymptotiques des groupes linéaires. Geom. Funct. Anal., 7(1):1–47, 1997.
  3. L. Bers. Spaces of Kleinian groups. In Several Complex Variables, I (Proc. Conf., Univ. of Maryland, College Park, Md., 1970), volume Vol. 155 of Lecture Notes in Math., pages 9–34. Springer, Berlin-New York, 1970.
  4. Anosov representations and dominated splittings. J. Eur. Math. Soc. (JEMS), 21(11):3343–3414, 2019.
  5. N. Bourbaki. Lie groups and Lie algebras. Chapters 4–6. pages xii+300, 2002. Translated from the 1968 French original by Andrew Pressley.
  6. The pressure metric for Anosov representations. Geom. Funct. Anal., 25(4):1089–1179, 2015.
  7. M. Bridson and A. Haefliger. Metric spaces of non-positive curvature, volume 319 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999.
  8. J.-P. Burelle and N. Treib. Schottky presentations of positive representations. Math. Ann., 382(3-4):1705–1744, 2022.
  9. M. Burger. Intersection, the Manhattan curve, and Patterson-Sullivan theory in rank 2222. Internat. Math. Res. Notices, (7):217–225, 1993.
  10. Entropy rigidity for cusped Hitchin representations. Preprint, arXiv:2201.04859, 2022.
  11. M. Chow and P. Sarkar. Local Mixing of One-Parameter Diagonal Flows on Anosov Homogeneous Spaces. Int. Math. Res. Not. IMRN, (18):15834–15895, 2023.
  12. M. Coornaert. Mesures de Patterson-Sullivan sur le bord d’un espace hyperbolique au sens de Gromov. Pacific J. Math., 159(2):241–270, 1993.
  13. K. Corlette and A. Iozzi. Limit sets of discrete groups of isometries of exotic hyperbolic spaces. Trans. Amer. Math. Soc., 351(4):1507–1530, 1999.
  14. S. Dey and M. Kapovich. Patterson-Sullivan theory for Anosov subgroups. Trans. Amer. Math. Soc., 375(12):8687–8737, 2022.
  15. Infinite volume and atoms at the bottom of the spectrum. Preprint, arXiv:2304.14565, 2023. To appear in Comptes Rendus Mathématique.
  16. S. Edwards and H. Oh. Temperedness of L2⁢(Γ∖G)superscript𝐿2Γ𝐺L^{2}(\Gamma\setminus G)italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Γ ∖ italic_G ) and positive eigenfunctions in higher rank. Comm. Amer. Math. Soc., 3:744–778, 2023.
  17. K. Falconer. Fractal geometry. John Wiley & Sons, Ltd., Chichester, third edition, 2014. Mathematical foundations and applications.
  18. Hausdorff dimension of limit sets for projective Anosov representations. J. Éc. polytech. Math., 10:1157–1193, 2023.
  19. Anosov representations and proper actions. Geom. Topol., 21(1):485–584, 2017.
  20. O. Guichard and A. Wienhard. Anosov representations: domains of discontinuity and applications. Invent. Math., 190(2):357–438, 2012.
  21. Harish-Chandra. Discrete series for semisimple Lie groups. II. Acta Math., 116:1–111, 1966.
  22. Y. Imayoshi and M. Taniguchi. An introduction to Teichmüller spaces. Springer-Verlag, Tokyo, 1992. Translated and revised from the Japanese by the authors.
  23. Anosov subgroups: dynamical and geometric characterizations. Eur. J. Math., 3(4):808–898, 2017.
  24. A Morse lemma for quasigeodesics in symmetric spaces and Euclidean buildings. Geom. Topol., 22(7):3827–3923, 2018.
  25. Hausdorff dimension of directional limit sets for self-joinings of hyperbolic manifolds. J. Mod. Dyn., 19:433–453, 2023.
  26. Tent property of the growth indicator functions and applications. Geom. Dedicata, 218(1):Paper No.14, 18, 2024.
  27. Ergodic dichotomy for subspace flows in higher rank. Preprint arXiv:2310.19976, 2023.
  28. Properly discontinous actions, growth indicators and conformal measures for transverse subgroups. Preprint arXiv:2306.06846, 2023.
  29. F. Labourie. Anosov flows, surface groups and curves in projective space. Invent. Math., 165(1):51–114, 2006.
  30. F. Ledrappier and P. Lessa. Dimension gap and variational principle for Anosov representations. Preprint arXiv:2310.13465, 2023.
  31. M. Lee and H. Oh. Dichotomy and measures on limit sets of Anosov groups. Preprint, arXiv:2203.06794, 2022. To appear in IMRN.
  32. M. Lee and H. Oh. Invariant Measures for Horospherical Actions and Anosov Groups. Int. Math. Res. Not. IMRN, (19):16226–16295, 2023.
  33. On the dimension of limit sets on ℙ⁢(ℝ3)ℙsuperscriptℝ3\mathbb{P}(\mathbb{R}^{3})blackboard_P ( blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) via stationary measures: the theory and applications. Preprint arXiv:2311.10265, 2023.
  34. Conformal dimension, volume 54 of University Lecture Series. American Mathematical Society, Providence, RI, 2010. Theory and application.
  35. A. Marden. The geometry of finitely generated kleinian groups. Ann. of Math. (2), 99:383–462, 1974.
  36. S. Patterson. The limit set of a Fuchsian group. Acta Math., 136(3-4):241–273, 1976.
  37. R. Potrie and A. Sambarino. Eigenvalues and entropy of a Hitchin representation. Invent. Math., 209(3):885–925, 2017.
  38. Conformality for a robust class of non-conformal attractors. J. Reine Angew. Math., 774:1–51, 2021.
  39. Anosov representations with Lipschitz limit set. Geom. Topol., 27(8):3303–3360, 2023.
  40. J.-F. Quint. Divergence exponentielle des sous-groupes discrets en rang supérieur. Comment. Math. Helv., 77(3):563–608, 2002.
  41. J.-F. Quint. Mesures de Patterson-Sullivan en rang supérieur. Geom. Funct. Anal., 12(4):776–809, 2002.
  42. J.-F. Quint. L’indicateur de croissance des groupes de Schottky. Ergodic Theory Dynam. Systems, 23(1):249–272, 2003.
  43. A. Sambarino. Hyperconvex representations and exponential growth. Ergodic Theory Dynam. Systems, 34(3):986–1010, 2014.
  44. A. Sambarino. A report on an ergodic dichotomy. Ergodic Theory Dynam. Systems, 44(1):236–289, 2024.
  45. D. Sullivan. The density at infinity of a discrete group of hyperbolic motions. Inst. Hautes Études Sci. Publ. Math., (50):171–202, 1979.
  46. D. Sullivan. Related aspects of positivity in Riemannian geometry. J. Differential Geom., 25(3):327–351, 1987.
  47. J. Tits. Classification of algebraic semisimple groups. Proc. Sympos. Pure. Math., 9:33–62, 1966.
  48. J. Tits. Représentations linéaires irréductibles d’un groupe réductif sur un corps quelconque. J. Reine Angew. Math., 247:196–220, 1971.
  49. T. Weich and L. Wolf. Absence of principal eigenvalues for higher rank locally symmetric spaces. Comm. Math. Phys., 403(3):1275–1295, 2023.
  50. L. Wolf and H.-W. Zhang. L2superscript𝐿2{L}^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-spectrum, growth indicator function and critical exponent on locally symmetric spaces. Preprint, arXiv:2311.11770, 2023.
  51. C. Yue. The ergodic theory of discrete isometry groups on manifolds of variable negative curvature. Trans. Amer. Math. Soc., 348(12):4965–5005, 1996.
Citations (4)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets