Collections of Orbits of Hyperplane Type in Homogeneous Spaces, Homogeneous Dynamics, and Hyperkähler Geometry

@article{Amerik2016CollectionsOO,
  title={Collections of Orbits of Hyperplane Type in Homogeneous Spaces, Homogeneous Dynamics, and Hyperk{\"a}hler Geometry},
  author={Ekaterina Amerik and Misha Verbitsky},
  journal={International Mathematics Research Notices},
  year={2016},
  volume={2020},
  pages={25-38}
}
Consider the space M = O(p, q)/O(p) × O(q) of positive p-dimensional subspaces in a pseudo-Euclidean space V of signature (p, q), where p > 0, q > 1 and (p, q) 6= (1, 2), with integral structure: V = VZ ⊗ R. Let Γ be an arithmetic subgroup in G = O(VZ), and R ⊂ VZ a Γ-invariant set of vectors with negative square. Denote by R the set of all positive p-planes W ⊂ V such that the orthogonal complement W contains r ∈ R. We prove that either R is dense in M or Γ acts on R with finitely many orbits… 

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