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Kapovich, Michael; Leeb, Bernhard and Porti, Joan (2018): Dynamics on flag manifolds: domains of proper discontinuity and cocompactness. In: Geometry & Topology, Vol. 22, No. 1: pp. 157-234

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Abstract

For noncompact semisimple Lie groups G with finite center, we study the dynamics of the actions of their discrete subgroups Gamma < 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 Gamma-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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