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Dynamics on flag manifolds: domains of proper discontinuity and cocompactness

Michael Kapovich, Bernhard Leeb and Joan Porti

Geometry & Topology 22 (2018) 157–234

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  GP. 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 Γ–action on various domains of proper discontinuity, in particular on domains in the full flag manifold  GB. 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.

Anosov subgroups, properly discontinuous actions, cocompact actions
Mathematical Subject Classification 2010
Primary: 53C35
Secondary: 22E40, 37B05, 51E24
Received: 11 October 2015
Revised: 11 May 2017
Accepted: 15 May 2017
Published: 31 October 2017
Proposed: Bruce Kleiner
Seconded: Anna Wienhard, Dmitri Burago
Michael Kapovich
Department of Mathematics
University of California
Davis, CA
United States
Bernhard Leeb
Mathematisches Institut
Universität München
Joan Porti
Departament de Matemàtiques
Universitat Autònoma de Barcelona