Nondivergence of Reductive group action on Homogeneous Spaces

Abstract: Let $G/\Gamma$ be the quotient of a semisimple Lie group $G$ by its arithmetic lattice. Let $H$ be a reductive algebraic subgroup of $G$ acting on $G/\Gamma$. The question we are interested in is whether there is a compact set of $G/\Gamma$ that intersects every H-orbit. We show that the failure of this can be explained by a single algebraic reason, which generalizes several previous results towards this question. Indeed, a more general result was obtained: when $H$ is a closed subgroup of $G$ satisfying $H=AM$ with $M$ semisimple, and $A$ an $\mathbb{R}$-diagonalizable subgroup contained in the centralizer of $M$ in $G$, then the failure of every $H$-orbit intersecting nontrivially with a fixed compact set is caused by finitely many algebraic obstructions.