Dynamics on flag manifolds: Domains of proper discontinuity and cocompactness

© 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.