IMBEDDINGS OF THREE-MANIFOLD GROUPS

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 π1π1-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 KK yields lens (or “lens-like”) spaces and how this relates to the knot subgroup structure of π1(S3−K)π1(S3−K). The authors use the formulation of a deformation theorem for π1π1-injective maps between certain kinds of Haken manifolds and develop some algebraic tools.