  We generalize [Vav] to give sufficient conditions, primarily on coarse
geometry, to ensure that a subset of a Cayley graph is a finite Hausdorff
distance from a subgroup. Using this result, we prove a partial converse to the
Flat Torus Theorem for CAT(0) groups. Also using this result, we give
sufficient conditions for subgroups and splittings to be invariant under
quasi-isometries.
