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 nitely generated subgroup H is the intersection of nite index subgroups of G. The rst 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 and link groups which are not subgroup separable. In section 1, the example K of BKS] is used to show that a closely related group L is not subgroup separable. L is the fundamental group of the complement of a very simple link: the chain of four circles illustrated in gure 2. In section 2, L plays the role of a poisonous subgroup in the proof of theorem 2.1 which characterizes the subgroup separability of certain amalgamated free products. For instance, as a special case of theorem 2.1 we obtain, is subgroup separable if and only if G is virtually F q Z for some q. The example of BKS] was exploited in LN] to produce various other non-subgroup separable groups. For instance, LN] construct a non-subgroup separable example R Z R, where R is the (4; 4; 2) triangle group. Their examples motivated theorem 2.1, which places their examples in a more general class. In section 3, theorem 2.1 is used to show that various knot groups are not subgroup separable. Perhaps the simplest such example is,