Nondivergence of Reductive group action on Homogeneous Spaces

Let $X=G\mathrm{/}\mathrm{\Gamma }$ 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 $\mathit{\text{every}}$ $H$-orbit. This generalizes previous results concerning certain special reductive group action on $X$ in this setting. When $G$ is of real rank one, $\mathrm{\Gamma }$ 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 $\mathit{\text{every}}$ $H$-orbit to a single compact set in $X$. This complements our results in the case of arithmetic lattice.