Dynamics on flag manifolds: domains of proper discontinuity and cocompactness

For noncompact semisimple Lie groups $G$ 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 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 $G/B$. We show in the regular case (of $B$-Anosov subgroups) that the latter domains are always nonempty if if $G$ has (locally) at least one noncompact simple factor not of the type $A_1, B_2$ or $G_2$.