Dynamics on flag manifolds: domains of proper discontinuity and cocompactness

Kapovich, Michael¹ ; Leeb, Bernhard² ; Porti, Joan³

¹ Department of Mathematics, University of California, Davis, CA, United States
² Mathematisches Institut, Universität München, München, Germany
³ Departament de Matemàtiques, Universitat Autònoma de Barcelona, Barcelona, Spain

Geometry & topology, Tome 22 (2018) no. 1, pp. 157-234.

Résumé

For noncompact semisimple Lie groups $G$ with finite center, 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, already made in the deep work of Benoist, 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  $G∕B$. In the regular case (eg of $B$–Anosov subgroups), we prove the nonemptiness of such domains if $G$ has (locally) at least one noncompact simple factor not of the type $A_1$, $B_2$ or $G_2$ by showing the nonexistence of certain ball packings of the visual boundary.

Détail

DOI : 10.2140/gt.2018.22.157

Classification : 53C35, 22E40, 37B05, 51E24
Keywords: Anosov subgroups, properly discontinuous actions, cocompact actions

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