Abstract Let X=G/ 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 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, 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 every H -orbit to a single compact set in X. This complements our results in the case of an arithmetic lattice.
ZHANG et al. (Fri,) studied this question.