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Bounded orbits on homogeneous spaces and bounded geodesics on locally symmetric spaces

  • Chengyang Wu
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Abstract

We present a unified approach to proving the following two results: (i) Let G be a Lie group, Γ a lattice in G, X=G/Γ, and let F be a one-parameter Ad-diagonalizable subgroup of G. Then the set {x∈ X:Fx is bounded in X} is hyperplane absolute winning on X. This fully solves a conjecture [Conjecture 7.1]{AGK} of An-Guan-Kleinbock in 2015. (ii) Let G be a connected semisimple Lie group with finite center and without compact factors, K≤ G a maximal compact subgroup, Γ≤ G a torsion-free lattice, and M=Kbackslash G/Γ. Then for any p∈ M, the set {ξ∈ SpM:γ(p,ξ) is bounded in M} is hyperplane absolute winning on SpM, where γ(p,ξ) denotes the geodesic ray through the base point p in the direction ξ. This fully solves a conjecture [Conjecture 1.2]{GW26} of Guan-Wu in 2026.

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Cite

BibTeXRIS
BibTeX
@misc{wu2026bounded,
  title = {{Bounded orbits on homogeneous spaces and bounded geodesics on locally symmetric spaces}},
  author = {Chengyang Wu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.24979},
  eprint = {2610.24979},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.24979},
}
RIS
TY  - UNPB
TI  - Bounded orbits on homogeneous spaces and bounded geodesics on locally symmetric spaces
AU  - Chengyang Wu
PY  - 2026
DA  - 2026/10/18
PB  - arXiv
DO  - 10.48550/arxiv.2610.24979
UR  - https://arxiv.org/abs/2610.24979
AB  - We present a unified approach to proving the following two results: (i) Let G be a Lie group, Γ a lattice in G, X=G/Γ, and let F be a one-parameter Ad-diagonalizable subgroup of G. Then the set {x∈ X:Fx is bounded in X} is hyperplane absolute winning on X. This fully solves a conjecture [Conjecture 7.1]{AGK} of An-Guan-Kleinbock in 2015. (ii) Let G be a connected semisimple Lie group with finite center and without compact factors, K≤ G a maximal compact subgroup, Γ≤ G a torsion-free lattice, and M=Kbackslash G/Γ. Then for any p∈ M, the set {ξ∈ S_pM:γ(p,ξ) is bounded in M} is hyperplane absolute winning on S_pM, where γ(p,ξ) denotes the geodesic ray through the base point p in the direction ξ. This fully solves a conjecture [Conjecture 1.2]{GW26} of Guan-Wu in 2026.
ER  -