Dynamics on flag manifolds: Domains of proper discontinuity and cocompactness

Michael Kapovich, Bernhard Leeb, Joan Porti

Research output: Contribution to journalArticleResearchpeer-review

32 Citations (Scopus)


© 2018, Mathematical Sciences Publishers. All rights reserved. For noncompact semisimple Lie groups G with finite center, we study the dynamics of the actions of their discrete subgroups Γ < G on the associated partial flag manifolds G/P. Our study is based on the observation, already made in the deep work of Benoist, that they exhibit also in higher rank a certain form of convergence-type dynamics. We identify geometrically domains of proper discontinuity in all partial flag manifolds. Under certain dynamical assumptions equivalent to the Anosov subgroup condition, we establish the cocompactness of the T-action on various domains of proper discontinuity, in particular on domains in the full flag manifold G/B. In the regular case (eg of B-Anosov subgroups), we prove the nonemptiness of such domains if G has (locally) at least one noncompact simple factor not of the type A1, B2 or G2 by showing the nonexistence of certain ball packings of the visual boundary.
Original languageEnglish
Pages (from-to)157-234
JournalGeometry and Topology
Issue number1
Publication statusPublished - 1 Jan 2018


  • Anosov subgroups
  • Cocompact actions
  • Properly discontinuous actions


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