Two-Bridge Knots have Residually Finite Groups

Abstract

If C is a property of groups, then we say that a group is residually-C if the intersection of all its normal subgroups with quotient group possessing the property C is the identity subgroup. The residual finiteness of the groups of fibred knots or Neuwirth knots, that is, those knot groups with finitely generated and therefore free commutator subgroup, has been known for some time [5, p. 63]. It was shown in [4] that certain knots with infinitely generated commutator subgroup share with Neuwirth knots the property that their commutator subgroup is residually a finite p-group, therefore implying the groups of these knots are residually finite. In this paper we show that the class of knot groups with commutator subgroups which are residually a finite p-group includes the groups of all two-bridge knots [7]. We take this as support for the conjecture that the class includes the groups of all alternating knots. The proof depends on the one-relator structure of a two-bridge knot group and proceeds as in [4] by adjoining a countable sequence of roots to a free group, employing theorems of Baumslag.