Dynamics on flag manifolds: domains of proper discontinuity and cocompactness

TL;DR

This paper investigates the dynamics of discrete subgroups acting on flag manifolds of noncompact semisimple Lie groups, identifying domains of proper discontinuity and establishing cocompactness under Anosov conditions, with implications for higher rank dynamics.

Contribution

It introduces a geometric framework for domains of proper discontinuity and proves cocompactness of group actions on these domains for Anosov subgroups in higher rank settings.

Findings

Identified geometrically domains of proper discontinuity in all partial flag manifolds.

Proved cocompactness of the action on these domains under Anosov subgroup conditions.

Showed nonemptiness of these domains in cases where the group has certain simple factors.

Abstract

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 $G_2$.