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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2209.06463 |
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| _version_ | 1866912832293961728 |
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| author | Zhang, Han Zhang, Runlin |
| author_facet | Zhang, Han Zhang, Runlin |
| contents | Let $X=G/Γ$ be the quotient of a semisimple Lie group $G$ by its non-cocompact arithmetic lattice. Let $H$ be a reductive algebraic subgroup of $G$ acting on $X$. We give several equivalent algebraic conditions on $H$ for the existence of a fixed compact set in $X$ intersecting \textit{every} $H$-orbit. This generalizes previous results concerning certain special reductive group action on $X$ in this setting.
When $G$ is of real rank one, $Γ$ is a non-cocompact lattice of $G$ and $H<G$ is an algebraic group, we also obtain an algebraic condition on $H$ which is equivalent to the return of \textit{every} $H$-orbit to a single compact set in $X$. This complements our results in the case of arithmetic lattice. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_06463 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Nondivergence of Reductive group action on Homogeneous Spaces Zhang, Han Zhang, Runlin Dynamical Systems 37B05, 22F30 Let $X=G/Γ$ be the quotient of a semisimple Lie group $G$ by its non-cocompact arithmetic lattice. Let $H$ be a reductive algebraic subgroup of $G$ acting on $X$. We give several equivalent algebraic conditions on $H$ for the existence of a fixed compact set in $X$ intersecting \textit{every} $H$-orbit. This generalizes previous results concerning certain special reductive group action on $X$ in this setting. When $G$ is of real rank one, $Γ$ is a non-cocompact lattice of $G$ and $H<G$ is an algebraic group, we also obtain an algebraic condition on $H$ which is equivalent to the return of \textit{every} $H$-orbit to a single compact set in $X$. This complements our results in the case of arithmetic lattice. |
| title | Nondivergence of Reductive group action on Homogeneous Spaces |
| topic | Dynamical Systems 37B05, 22F30 |
| url | https://arxiv.org/abs/2209.06463 |