Rigidity of invariant convex sets in symmetric spaces

By Bruce Kleiner, B. Leeb. 2004

62 citations recorded by the source corpus.

The main result implies that a proper convex subset of an irreducible higher rank symmetric space cannot have Zariski dense stabilizer.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cbfeb8a7d3f751e09cffb587785747e1239e0a24

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Anosov subgroups: dynamical and geometric characterizations (2017, background): “Convex cocompactness turns out to bemuch too restrictive a condition: it was shown byKleiner and the second author [24] that in higher rank only few subgroups are convex cocompact.”
Finsler bordifications of symmetric and certain locally symmetric spaces (2015, background): “We note that this list of equivalences is nearly a perfect match to the list of equivalent definitions of convex cocompact subgroups of rank-1 Lie groups (see [27]), except that convex-cocompactness is (by necessity) missing; see [38].”
Anosov representations: domains of discontinuity and applications (2011, background): “Anosov representations into Lie groups of rank one are exactly convex cocompact representations (see Theorem 1.8) For Lie groups G of real rank ≥ 2, a rigidity result of Kleiner-Leeb [52] and Quint [65] says that the class of convex cocompact subgroups of G reduces to products of uniform lattices an”
Convex cocompactness in pseudo-Riemannian hyperbolic spaces (2017, result): “This naive generalization of convex cocompactness turns out to be quite restrictive due to the following general result proved independently by Kleiner–Leeb [36] and Quint [43].”
Convex cocompact actions in real projective geometry (2017, background): “ntheotherhand,inhigher-ranksemisimpleLiegroupsG,thecondition thatadiscretesubgroup actcocompactlyonsomenonemptyconvexsubset oftheRiemanniansymmetricspace G=Kturnsouttobequiterestrictive: Kleiner–Leeb [KL] and Quint [Q] proved, for example, that if Gis simple and such a subgroup is Zariski-dense in G”
MAPPING CLASS GROUP DYNAMICS ON SURFACE GROUP REPRESENTATIONS (2005, background): “ρ] ∈ Ω, then the action of ModΣ on Ω is proper. This gives a general analytic condition implying properness. Unfortunately, convex cocompactness is extremely restrictive; Kleiner and Leeb have proved [66] that in rank > 1 such groups are never Zariski dense. However, we know many examples (the de”