Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-06-08T06:41:20.551Z Has data issue: false hasContentIssue false
  • English
  • Français

STRUCTURAL STABILITY OF MEANDERING-HYPERBOLIC GROUP ACTIONS

Part of: Complex dynamical systems Lie groups Special aspects of infinite or finite groups

Published online by Cambridge University Press:  27 December 2022

Michael Kapovich ,
Sungwoon Kim  and
Jaejeong Lee Open the ORCID record for Jaejeong Lee [Opens in a new window]
Michael Kapovich
Affiliation:
Department of Mathematics, University of California, 1 Shields Ave., Davis, CA 95616, USA (kapovich@math.ucdavis.edu)
Sungwoon Kim
Affiliation:
Department of Mathematics, Jeju National University, Jeju 63243, Republic of Korea (sungwoon@jejunu.ac.kr)
Jaejeong Lee*
Affiliation:
Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea
*
(jaejeong@snu.ac.kr)
Get access Rights & Permissions [Opens in a new window]

Abstract

In his 1985 paper, Sullivan sketched a proof of his structural stability theorem for differentiable group actions satisfying certain expansion-hyperbolicity axioms. In this paper, we relax Sullivan’s axioms and introduce a notion of meandering hyperbolicity for group actions on geodesic metric spaces. This generalization is substantial enough to encompass actions of certain nonhyperbolic groups, such as actions of uniform lattices in semisimple Lie groups on flag manifolds. At the same time, our notion is sufficiently robust, and we prove that meandering-hyperbolic actions are still structurally stable. We also prove some basic results on meandering-hyperbolic actions and give other examples of such actions.

MSC classification

Primary: 37F15: Expanding maps; hyperbolicity; structural stability
Secondary: 20F67: Hyperbolic groups and nonpositively curved groups 22E40: Discrete subgroups of Lie groups
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 23 , Issue 2 , March 2024 , pp. 753 - 810
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ballmann, W., Gromov, M. and Schroeder, V., Manifolds of Nonpositive Curvature , Progress in Mathematics , vol. 61 (Birkhäuser Boston, Inc., Boston, MA, 1985).Google Scholar
Barbot, T., ‘Three-dimensional Anosov flag manifolds’, Geom. Topol. 14(1) (2010), 153–191.CrossRefGoogle Scholar
Bochi, J., Potrie, R. and Sambarino, A., ‘Anosov representations and dominated splittings’, J. Eur. Math. Soc. (JEMS) 21(11) (2019), 3343–3414.CrossRefGoogle Scholar
Bowditch, B. H., ‘Geometrical finiteness for hyperbolic groups’, J. Funct. Anal. 113(2) (1993), 245–317.CrossRefGoogle Scholar
Bowditch, B. H., ‘Spaces of geometrically finite representations’, Ann. Acad. Sci. Fenn. Math. 23(2) (1998), 389–414.Google Scholar
Bowditch, B. H., ‘A topological characterisation of hyperbolic groups’, J. Amer. Math. Soc. 11(3) 1998, 643–667.CrossRefGoogle Scholar
Bridson, M. R. and Haeiger, A., Metric Spaces of Non-Positive Curvature , Grundlehren der Mathematischen Wissenschaften , vol. 319 (Springer-Verlag, Berlin, 1999).Google Scholar
Canary, R. D., Epstein, D. B. A., and Green, P., Notes on Notes of Thurston , London Math. Soc. Lecture Note Ser. vol. 111 (Cambridge Univ. Press, Cambridge, 1987), 3–92.Google Scholar
Choi, S. and Goldman, W., ‘Topological tameness of Margulis spacetimes’, Amer. J. Math. 139(2) (2017), 297–345.CrossRefGoogle Scholar
Coornaert, M., ‘Mesures de Patterson–Sullivan sur le bord d’un espace hyperbolique au sens de Gromov’, Pacific J. Math. 159(2) (1993), 241–270.CrossRefGoogle Scholar
Coornaert, M. and Papadopoulos, A., Symbolic Dynamics and Hyperbolic Groups, Lecture Notes in Mathematics, vol. 1539 (Springer-Verlag, Berlin, 1993).Google Scholar
Corlette, K., ‘Hausdorff dimensions of limit sets. I’, Invent. Math. 102(3) (1990), 521–541.CrossRefGoogle Scholar
Danciger, J., Guéritaud, F. and Kassel, F., ‘Convex cocompact actions in real projective geometry’, Preprint, 2017, arXiv:1704.08711.Google Scholar
Deroin, B., ‘Locally discrete expanding groups of analytic diffeomorphisms of the circle’, J. Topol. 13(3) (2020), 1216–1229.CrossRefGoogle Scholar
Druţu, C. and Kapovich, M., Geometric Group Theory , American Mathematical Society Colloquium Publications , vol. 63 (American Mathematical Society, Providence, RI, 2018).Google Scholar
Eberlein, P. B., Geometry of Nonpositively Curved Manifolds, Chicago Lectures in Mathematics (University of Chicago Press, Chicago, IL, 1996).Google Scholar
Freden, E. M., ‘Negatively curved groups have the convergence property. I’, Ann. Acad. Sci. Fenn. Ser. A I Math. 20(2) (1995), 333–348.Google Scholar
Gromov, M., Hyperbolic Groups , Essays in Group Theory, Math. Sci. Res. Inst. Publ . vol. 8 (Springer, New York, 1987), 75–263.CrossRefGoogle Scholar
Guichard, O. and Wienhard, A., ‘Anosov representations: Domains of discontinuity and applications’, Invent. Math. 190(2) (2012), 357–438.CrossRefGoogle Scholar
Izeki, H., Quasiconformal stability of Kleinian groups and an embedding of a space of flat conformal structures, Conform. Geom. Dyn. 4 (2000), 108–119.CrossRefGoogle Scholar
Kapovich, M., Hyperbolic Manifolds and Discrete Groups , Progress in Mathematics, vol. 183 (Birkhäuser Boston, Inc., Boston, MA, 2001).Google Scholar
Kapovich, M. and Leeb, B., Discrete Isometry Groups of Symmetric Spaces , Handbook of Group Actions. vol. IV, Adv. Lect. Math. (ALM) , vol. 41 (Int. Press, Somerville, MA, 2018), 191–290.Google Scholar
Kapovich, M. and Leeb, B., ‘Finsler bordifications of symmetric and certain locally symmetric spaces’, Geom. Topol. 22(5) (2018), 2533–2646.CrossRefGoogle Scholar
Kapovich, M., Leeb, B. and Porti, J., ‘Morse actions of discrete groups on symmetric space’, Preprint, 2014, arXiv:1403.7671.Google Scholar
Kapovich, M., Leeb, B. and Porti, J., ‘Some recent results on Anosov representations’, Transform. Groups 21(4) (2016), 1105–1121.CrossRefGoogle Scholar
Kapovich, M., Leeb, B. and Porti, J., ‘Anosov subgroups: Dynamical and geometric characterizations’, Eur. J. Math. 3(4) (2017), 808–898.CrossRefGoogle Scholar
Kapovich, M., Leeb, B. and Porti, J., ‘A Morse lemma for quasigeodesics in symmetric spaces and Euclidean buildings’, Geom. Topol. 22(7) (2018), 3827–3923.CrossRefGoogle Scholar
Labourie, F., ‘Anosov flows, surface groups and curves in projective space’, Invent. Math. 165(1) (2006), 51–114.CrossRefGoogle Scholar
Mann, K. and Manning, J. F., ‘Stability for hyperbolic groups acting on boundary spheres’, Preprint, 2021, arXiv:2104.01269.Google Scholar
Marden, A., ‘The geometry of finitely generated Kleinian groups’, Ann. of Math. (2) 99 (1974), 383–462.CrossRefGoogle Scholar
Maskit, B., Kleinian Groups, Grundlehren der Mathematischen Wissenschaften , vol. 287 (Springer-Verlag, Berlin, 1988).Google Scholar
Matsuzaki, K., ‘Structural stability of Kleinian groups’, Michigan Math. J. 44(1) (1997), 21–36.CrossRefGoogle Scholar
Series, C., ‘Symbolic dynamics for geodesic flows’, Acta Math. 146(1–2)(1981), 103–128.CrossRefGoogle Scholar
Stecker, F. and Treib, N., ‘Domains of discontinuity in oriented flag manifolds’, Preprint, 2018, arXiv:1806.04459.Google Scholar
Sullivan, D., ‘Quasiconformal homeomorphisms and dynamics. II. Structural stability implies hyperbolicity for Kleinian groups’, Acta Math. 155(3–4) (1985), 243–260.CrossRefGoogle Scholar
Tukia, P., ‘Convergence groups and Gromov’s metric hyperbolic spaces’, New Zealand J. Math. 23(2) (1994), 157–187.Google Scholar
Tukia, P., ‘Conical limit points and uniform convergence groups’, J. Reine Angew. Math. 501 (1998), 71–98.CrossRefGoogle Scholar
Yue, C., ‘Dimension and rigidity of quasi-Fuchsian representations’, Ann. of Math. 2, 143(2) (1996), 331–355.CrossRefGoogle Scholar