Dynamics on flag manifolds: domains of proper discontinuity and cocompactness

By Michael Kapovich, B. Leeb, J. Porti. 2013

93 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c3b5a40cdafc07ed0c7fd4b9a38f3e0f7c6c37d9

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Citations in the corpus

AN INVITATION TO HIGHER TEICHMÜLLER THEORY (2019, methodology): “This dynamical point of view has been first observed by Labourie [2006] and applied by Sambarino [2014a] and has been key in several interesting developments. Using the thermodynamical formalism Bridgeman, Canary, Labourie, and Sambarino [2015] define the pressure metric on the Hitchin component TH ”
Anosov subgroups: dynamical and geometric characterizations (2017, methodology): “Expansivity (ii) is used in [18,19] to establish the cocompactness of -actions on suitable domains of discontinuity in flag manifolds.”
An invitation to higher Teichm\"uller theory (2018, methodology): “In fact, we expect, that for any cocompact domain of discontintuity which is constructed through a balanced thickening in the sense of [72], the quotient”
Finsler bordifications of symmetric and certain locally symmetric spaces (2015, methodology): “The G–action on x X Fins is determined by (fills in) the G–action on @FuX Š Flag mod , and our approach is based on the study of the dynamics of the G– action on its associated flag manifolds in [34].”