Imbeddings of three-manifold groups

Francisco Gonzalez-Acuna, Wilbur C. Whitten9780821825341, 0821825348

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.