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arXiv · 2209.06463

Nondivergence of Reductive group action on Homogeneous Spaces

Han Zhang·Runlin Zhang
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Abstract

Let $X=G/\Gamma$ 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, $\Gamma$ 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.

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Han Zhang, Runlin Zhang. 2022-09-14. Nondivergence of Reductive group action on Homogeneous Spaces. https://arxiv.org/abs/2209.06463

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