Subgroup separability, knot groups and graph manifolds

This paper answers a question of Burns, Karrass and Solitar by giving examples of knot and link groups which are not subgroup-separable. For instance, it is shown that the fundamental group of the square knot complement is not subgroup separable. Let L L denote the fundamental group of the link consisting of a chain of 4 4 circles. It is shown that L L is not subgroup separable. Furthermore, it is shown that L L is a subgroup of every known non-subgroup separable compact 3-manifold group. It is asked whether all such examples contain L L .