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 $G_2$.

65 pages

References in corpus (5)

• Anosov representations: Domains of discontinuity and applications
• A Morse Lemma for quasigeodesics in symmetric spaces and euclidean buildings
• A characterization of irreducible symmetric spaces and Euclidean buildings of higher rank by their asymptotic geometry
• Morse actions of discrete groups on symmetric space
• Discrete isometry groups of symmetric spaces

Cited by in corpus (28)

• Finsler bordifications of symmetric and certain locally symmetric spaces
• The geometry of maximal components of the PSp(4,R) character variety
• A combination theorem for Anosov subgroups
• SO(p,q)-Higgs bundles and higher Teichmüller components
• Higgs bundles and geometric structures on manifolds
• Geometry of compact complex manifolds associated to generalized quasi-Fuchsian representations
• Regularity of limit sets of Anosov representations
• Lectures on complex hyperbolic Kleinian groups
• Schottky groups and maximal representations
• Margulis Multiverse: Infinitesimal Rigidity, Pressure Form and Convexity
• A general Cayley correspondence and higher Teichmüller spaces
• Morse actions of discrete groups on symmetric spaces: Local-to-global principle
• Weak topology on CAT(0) spaces
• Groupes convexes-cocompacts en rang supérieur
• Cubulated hyperbolic groups admit Anosov representations
• Birational Kleinian groups
• Rigidity of convex divisible domains in flag manifolds
• Projective Structures with (Quasi-)Hitchin Holonomy
• Equidistribution of expanding translates of curves and Diophantine approximation on matrices
• Some recent results on Anosov representations
• Nearly geodesic immersions and domains of discontinuity
• Anosov representations acting on homogeneous spaces: domains of discontinuity
• Uniformization of compact complex manifolds by Anosov representations
• Complete affine manifolds with Anosov holonomy groups
• Dimension gap and variational principle for Anosov representations
• Fiber bundles associated with Anosov representations
• Lyapunov exponents and stability properties of higher rank representations
• Actions on products of CAT(-1) spaces