#### Volume 22, issue 1 (2018)

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Dynamics on flag manifolds: domains of proper discontinuity and cocompactness

### Michael Kapovich, Bernhard Leeb and Joan Porti

Geometry & Topology 22 (2018) 157–234
##### Abstract

For noncompact semisimple Lie groups $G$ with finite center, we study the dynamics of the actions of their discrete subgroups $\Gamma 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 $\Gamma \phantom{\rule{0.3em}{0ex}}$–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.

##### Keywords
Anosov subgroups, properly discontinuous actions, cocompact actions
##### Mathematical Subject Classification 2010
Primary: 53C35
Secondary: 22E40, 37B05, 51E24