Knot Groups That Are Not Subgroup Separable

. 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 denote the fundamental group of the link consisting of a chain of 4 circles. It is shown that L is not subgroup separable. Furthermore, it is shown that L is a subgroup of every known non-subgroup separable compact 3-manifold group. It is asked whether all such examples contain L. A group G is said to be subgroup separable if each finitely generated subgroup H is the intersection of finite index subgroups of G. The first example of a 3-manifold group which is not subgroup separable was given in [BKS]. In that paper, Burns, Karrass, and Solitar observed that their example K is not a knot group, and they asked if knot groups are subgroup separable. The primary purpose of this paper is to answer their question by giving examples of knot..