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Let $G/\Gamma$ be the quotient of a semisimple Lie group $G$ by itsarithmetic 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 setof $G/\Gamma$ that intersects every H-orbit. We show that the failure of thiscan be explained by a single algebraic reason, which generalizes severalprevious 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 fixedcompact set is caused by finitely many algebraic obstructions.

Nondivergence of Reductive group action on Homogeneous Spaces