This work deals with the two broad questions of how three-manifold groups imbed in one another and how such imbeddings relate to any corresponding \(\pi _1\)-injective maps. The focus is on when a given three-manifold covers another given manifold. In particular, the authors are concerned with 1) determining which three-manifold groups are not cohopfian—that is, which three-manifold groups imbed properly in themselves; 2) finding the knot subgroups of a knot group; and 3) investigating when surgery on a knot \(K\) yields lens (or “lens-like”) spaces and how this relates to the knot subgroup structure of \(\pi _1(S^3-K)\). The authors use the formulation of a deformation theorem for \(\pi _1\)-injective maps between certain kinds of Haken manifolds and develop some algebraic tools.