Nondivergence of Reductive group action on Homogeneous Spaces

TL;DR

This paper establishes algebraic criteria for the existence of compact sets intersecting all orbits of reductive groups acting on homogeneous spaces, extending previous results and covering real rank one cases.

Contribution

It provides new algebraic conditions characterizing when reductive group actions on homogeneous spaces have bounded orbits, generalizing prior specific cases.

Findings

Algebraic conditions for fixed compact sets intersecting all orbits

Extension of results to real rank one cases

Characterization of orbit return properties in homogeneous spaces

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.