Advertisement

European Journal of Mathematics

, Volume 3, Issue 4, pp 808–898 | Cite as

Anosov subgroups: dynamical and geometric characterizations

  • Michael Kapovich
  • Bernhard Leeb
  • Joan Porti
Research Article

Abstract

We study infinite covolume discrete subgroups of higher rank semisimple Lie groups, motivated by understanding basic properties of Anosov subgroups from various viewpoints (geometric, coarse geometric and dynamical). The class of Anosov subgroups constitutes a natural generalization of convex cocompact subgroups of rank one Lie groups to higher rank. Our main goal is to give several new equivalent characterizations for this important class of discrete subgroups. Our characterizations capture “rank one behavior” of Anosov subgroups and are direct generalizations of rank one equivalents to convex cocompactness. Along the way, we considerably simplify the original definition, avoiding the geodesic flow. We also show that the Anosov condition can be relaxed further by requiring only non-uniform unbounded expansion along the (quasi)geodesics in the group.

Keywords

Discrete subgroups Anosov subgroups Symmetric spaces 

Mathematics Subject Classification

22E40 20F65 53C35 

References

  1. 1.
    Albuquerque, P.: Patterson–Sullivan theory in higher rank symmetric spaces. Geom. Funct. Anal. 9(1), 1–28 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ballmann, W., Gromov, M., Schroeder, V.: Manifolds of Nonpositive Curvature. Progress in Mathematics, vol. 61. Birkhäuser, Boston (1985)CrossRefzbMATHGoogle Scholar
  3. 3.
    Benoist, Y.: Propriétés asymptotiques des groupes linéaires. Geom. Funct. Anal. 7(1), 1–47 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bowditch, B.: A topological characterisation of hyperbolic groups. J. Amer. Math. Soc. 11(3), 643–667 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Champetier, C.: Petite simplification dans les groupes hyperboliques. Ann. Fac. Sci. Toulouse Math. (6) 3(2), 161–221 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Coornaert, M., Papadopoulos, A.: Symbolic Dynamics and Hyperbolic Groups. Lecture Notes in Mathematics, vol. 1539. Springer, Berlin (1993)zbMATHGoogle Scholar
  7. 7.
    Eberlein, P.B.: Geometry of Nonpositively Curved Manifolds. Chicago Lecture Notes in Mathematics. University of Chicago Press, Chicago (1997)Google Scholar
  8. 8.
    Frances, Ch.: Lorentzian Kleinian groups. Comment. Math. Helv. 80(4), 883–910 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Freden, E.M.: Negatively curved groups have the convergence property I. Ann. Acad. Sci. Fenn. Ser. A I Math. 20(2), 333–348 (1995)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Gromov, M.: Hyperbolic groups. In: Gersten, S.M. (ed.) Essays in Group Theory. Mathematical Sciences Research Institute Publications, vol. 8, pp. 75–263. Springer, New York (1987)Google Scholar
  11. 11.
    Guéritaud, F., Guichard, O., Kassel, F., Wienhard, A.: Anosov representations and proper actions (2015). arXiv:1502.03811
  12. 12.
    Guichard, O., Wienhard, A.: Anosov representations: domains of discontinuity and applications. Invent. Math. 190(2), 357–438 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Helgason, S.: Differential Geometry and Symmetric Spaces. Pure and Applied Mathematics, vol. 12. Academic Press, New York (1962)zbMATHGoogle Scholar
  14. 14.
    Jantzen, J.C.: Representations of Algebraic Groups. Mathematical Surveys and Monographs, vol. 107, 2nd edn. American Mathematical Society, Providence (2003)Google Scholar
  15. 15.
    Kapovich, M., Leeb, B.: Finsler bordifications of symmetric and certain locally symmetric spaces (2015). arXiv:1505.03593
  16. 16.
    Kapovich, M., Leeb, B.: Discrete isometry groups of symmetric spaces (2017). arXiv:1703.02160. Handbook of Group Actions, vol. 4 (to appear)
  17. 17.
    Kapovich, M., Leeb, B., Millson, J.: Convex functions on symmetric spaces, side lengths of polygons and the stability inequalities for weighted configurations at infinity. J. Differential Geom. 81(2), 297–354 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kapovich, M., Leeb, B., Porti, J.: Dynamics at infinity of regular discrete subgroups of isometries of higher rank symmetric spaces (2013). arXiv:1306.3837v1
  19. 19.
    Kapovich, M., Leeb, B., Porti, J.: Dynamics on flag manifolds: domains of proper discontinuity and cocompactness (2013). arXiv:1306.3837. Geom. Topol. (to appear)
  20. 20.
    Kapovich, M., Leeb, B., Porti, J.: Morse actions of discrete groups on symmetric spaces (2014). arXiv:1403.7671
  21. 21.
    Kapovich, M., Leeb, B., Porti, J.: A Morse Lemma for quasigeodesics in symmetric spaces and euclidean buildings (2014). arXiv:1411.4176
  22. 22.
    Kapovich, M., Leeb, B., Porti, J.: Some recent results on Anosov representations. Transform. Groups 21(4), 1105–1121 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kleiner, B., Leeb, B.: Rigidity of quasi-isometries for symmetric spaces and euclidean buildings. Inst. Hautes Études Sci. Publ. Math. 86, 115–197 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kleiner, B., Leeb, B.: Rigidity of invariant convex sets in symmetric spaces. Invent. Math. 163(3), 657–676 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Labourie, F.: Anosov flows, surface groups and curves in projective space. Invent. Math. 165(1), 51–114 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Leeb, B.: A Characterization of Irreducible Symmetric Spaces and Euclidean Buildings of Higher Rank by their Asymptotic Geometry. Bonner Mathematische Schriften, vol. 326. Universität Bonn, Bonn (2000). arXiv:0903.0584
  27. 27.
    Mineyev, I.: Flows and joins of metric spaces. Geom. Topol. 9(1), 403–482 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Parreau, A.: La distance vectorielle dans les immeubles affines et les espaces symétriques (in preparation)Google Scholar
  29. 29.
    Sullivan, D.: Quasiconformal homeomorphisms and dynamics. II. Structural stability implies hyperbolicity for Kleinian groups. Acta Math. 155(3–4), 243–260 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Tukia, P.: Convergence groups and Gromov’s metric hyperbolic spaces. New Zealand J. Math. 23(2), 157–187 (1994)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaDavisUSA
  2. 2.Mathematisches InstitutUniversität MünchenMunichGermany
  3. 3.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterraSpain

Personalised recommendations