© 1996, Topology Atlas

Recall that the exterior of a torus knot covers itself. Is this true for any knot other than a torus knot? In fact, does the group of any nontorus knot imbed in itself nontrivially with finite index? (The answer to each of these questions is, No.") We might begin by ask- ing a more general question:

How do 3-manifold groups imbed in one another and how do these imbeddings relate to any corresponding group-injective maps?

To focus our attention, a group that does not properly imbed in itself is cohopfian; hopficity and cohopficity are dual concepts. Cohopfian groups were considered by Baer [B] over fifty years ago and were reintroduced in [GW1] as another approach to the study of 3-manifold groups. Here are some known results.

1. Let M be a Haken 3-manifold, different from a collar, whose boundary is a nonempty union of incompressible tori. The group of M is cohopfian if and only if the collection of components of the characteristic submanifold of M meeting the boundary of M is a disjoint union of collars [GW1].
2. The group of a (tame) nontrivial knot K is cohopfian if and only if K is not a torus knot, a cable knot, or a composite knot [GW1].
3. A nontrivial knot whose group properly imbeds in itself with finite index is a torus knot [GW1].

Remark. As for hopficity, a nice question is: Does there exist a knot group whose commutator subgroup is not hopfian?

References

[B] R. Baer, Groups without proper isomorphic quotient groups," Bull. Amer. Math. Soc. 50 (1944), 267-278.
[GW1] F. Gonzalez-Acuna and W. Whitten, Imbeddings of Three-Manifold Groups, Mem. Amer. Math. Soc. 474 (1992).
[GW2] ________, Cohopficity of 3-manifold groups," Top. and its Appli. 56 (1994), 87-97.
[GLW] F. Gonzalez-Acuna, R. Litherland, and W. Whitten, Cohopficity of Seifert-bundle groups," Trans. Amer. Math. Soc. 341 (1994), 143-155.
[WW] S. Wang and Y.-Q. Wu, A covering invariant of graph manifolds," Proc. London Math. Soc., to appear.