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Dynamics on flag
manifolds: domains of proper discontinuity and
cocompactness
Michael Kapovich, Bernhard Leeb and Joan Porti |
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Geometry & Topology 22 (2018) 157–234
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Abstract |
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For noncompact semisimple Lie groups with finite center, we study the dynamics of the actions of their discrete subgroups on the associated partial flag manifolds . 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 –action on various domains of proper discontinuity, in particular on domains in the full flag manifold . In the regular case (eg of –Anosov subgroups), we prove the nonemptiness of such domains if has (locally) at least one noncompact simple factor not of the type , or by showing the nonexistence of certain ball packings of the visual boundary. |
Keywords
Anosov subgroups, properly discontinuous actions, cocompact
actions
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Mathematical Subject Classification 2010
Primary: 53C35
Secondary: 22E40, 37B05, 51E24
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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
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Authors |
| Michael Kapovich | |
| Department of Mathematics University of California Davis, CA United States |
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| Bernhard Leeb | |
| Mathematisches Institut Universität München München Germany |
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| Joan Porti | |
| Departament de Matemàtiques Universitat Autònoma de Barcelona Barcelona Spain |
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