Dynamics on flag manifolds: domains of proper discontinuity and cocompactness

Abstract

For noncompact semisimple Lie groups [math] with finite center, we study the dynamics of the actions of their discrete subgroups [math] on the associated partial flag manifolds [math] . 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 [math] –action on various domains of proper discontinuity, in particular on domains in the full flag manifold [math] . In the regular case (eg of [math] –Anosov subgroups), we prove the nonemptiness of such domains if [math] has (locally) at least one noncompact simple factor not of the type [math] , [math] or [math] by showing the nonexistence of certain ball packings of the visual boundary.