# 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…

## 8 Citations

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