Dynamics on flag manifolds: domains of proper discontinuity and cocompactness
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 $G_2$.
 Publication:

arXiv eprints
 Pub Date:
 June 2013
 arXiv:
 arXiv:1306.3837
 Bibcode:
 2013arXiv1306.3837K
 Keywords:

 Mathematics  Metric Geometry;
 Mathematics  Dynamical Systems;
 Mathematics  Group Theory;
 22E40;
 53C35;
 37B05;
 51E24
 EPrint:
 65 pages