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


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.


Discrete subgroups Anosov subgroups Symmetric spaces 

Mathematics Subject Classification

22E40 20F65 53C35 


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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

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