Subgroup Separability of the Figure 8 Knot Group
by
Daniel Wise
Cornell

A subgroup H of a group G is said to be separable provided that H is the intersection of finite index subgroups of G. A group is called subgroup separable provided that every finitely generated subgroup is separable. Well known examples of groups which are subgroup separable include free groups (M. Hall) and surface groups (P. Scott).

Important work building on these earlier results has been done by Brunner-Burns-Solitar, Gitik, Long, Niblo, Tretkoff and others. These authors obtain many examples of subgroup separable 3-manifold groups. However, an example of a compact 3-manifold group which is not subgroup separable was given by Burns-Karrass-Solitar. Other examples were given more recently by Rubinstein-Wang.

I have proven that every geometrically finite subgroup of the figure 8 knot group is separable. The same proof works for many other hyperbolic 3-manifold groups.

One reason why low dimensional topologists are interested in subgroup separability is because it allows certain immersions to lift to embeddings in a finite cover. For instance, as an application of the result, we find that if M is the figure 8 knot complement, then every properly immersed incompressible surface S in M, of minimal self- intersection, lifts to an embedding in a finite cover of M.

http://math.cornell.edu/~daniwise

Date received: March 9, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caca-64.