On residually finite knot groups

The residual finiteness of the class of groups of fibred knots, or those knot groups with finitely generated and, therefore, free commutator subgroups, has been known for some time. Using Baumslag's results on absolutely parafree groups, this paper extends the result to twist knots (Whitehead doubles of the trivial knot) and certain other classes of nonfibred knots whose minimal spanning surface has complement with free fundamental group. As a by-product more explicit finite representations, namely cyclic extensions of certain p-groups, are obtained for these knot groups and the groups of fibred knots. Finally composites of two such knots also have residually finite groups. 0. Introduction. A group is residually finite if the intersection of its normal subgroups of finite index is the identity. The residual finiteness of the groups of fibred knots, or those knot groups with finitely generated and, therefore, free commutator subgroups, has been known for some time [13, p. 63]. We include in §2 a modification of the proof of this theorem which gives more explicit information on the structure of a sufficient set of finite homomorphic images. This result is extended, using Baumslag's theorems on parafree groups ([1] and [2]), to those knot groups whose commutator subgroup is an ascending union of parafree subgroups and, therefore, in particular to those knot groups whose commutator subgroup can be built up from a free group by the careful iterated adjunction of a countable sequence of roots. In §3 we give a brief account of Neuwirth's analysis of the structure of the commutator subgroup of a knot group, enabling us to give in §4 some examples of such residually finite knot groups including the groups of the so-called twist knots, or Whitehead doubles of the trivial knot. We also show that any composite of such knots has a residually finite group. Finally, we give a brief indication of how this work might be continued. We remark that P. Stebe has shown that hose knots have residually finite groups by showing their groups are nc [16], that is for each two elements g± and g2 in the group, either gx = g2 or else there is a normal subgroup N of finite index such that Si^St. m°d N for each integer n. Presented to the Society, January 18, 1972; received by the editors May 25, 1971. AMS 1970 subject classifications. Primary 55A25, 20E25, 55A05; Secondary 20E10. C1) Acknowledgements. This work is essentially the author's doctoral thesis at the University of Wisconsin and I gratefully acknowledge the guidance, assistance, and encouragement of my advisor, Professor Joseph Martin, in its preparation. I would also like to thank Professor Richard Brück for many helpful conversations. Copyright © 1972, American Mathematical Society