Dynamics on flag manifolds: domains of proper discontinuity and cocompactness

Kapovich, Michael; Leeb, Bernhard; Porti, Joan

doi:10.2140/gt.2018.22.157

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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

DOI: 10.2140/gt.2018.22.157

Abstract

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 $Γ$ –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 $A_{1}$ , $B_{2}$ or $G_{2}$ by showing the nonexistence of certain ball packings of the visual boundary.

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Keywords

Anosov subgroups, properly discontinuous actions, cocompact actions

Mathematical Subject Classification 2010

Primary: 53C35

Secondary: 22E40, 37B05, 51E24

References

Publication

Received: 11 October 2015

Revised: 11 May 2017

Accepted: 15 May 2017

Published: 31 October 2017

Proposed: Bruce Kleiner

Seconded: Anna Wienhard, Dmitri Burago

Authors

Michael Kapovich
Department of Mathematics University of California Davis, CA United States

Bernhard Leeb
Mathematisches Institut Universität München München Germany

Joan Porti
Departament de Matemàtiques Universitat Autònoma de Barcelona Barcelona Spain

Volume 22, issue 1 (2018)