Dynamics on flag manifolds: domains of proper discontinuity and cocompactness

2013/06/30 by Michael Kapovich, Bernhard Leeb, Joan Porti · 2 citations
Mathematics · #math.MG #math.DS #math.GR #msc:22E40 #msc:53C35 #msc:37B05 #msc:51E24

paper · pdf · doi:10.2140/gt.2018.22.157

published as Geom. Topol. 22 (2018) 157-234 · 65 pages

arxiv created 2017/03/05 · arxiv updated 2018/03/16

Abstract

For noncompact semisimple Lie groups G we study the dynamics of the actions of their discrete subgroups Γ<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 Γ-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 A₁, B₂ or G₂.

Cited by

• Balanced ideals and domains of discontinuity of Anosov representations
• Geometry of compact complex manifolds associated to generalized quasi-Fuchsian representations

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