The Groups of Fibred 2-knots

We explore algebraic characterizations of 2-knots whose associated knot manifolds fibre over lower-dimensional orbifolds, and consider also some issues related to the groups of higherdimensional fibred knots. Nontrivial classical knot groups have cohomological dimension 2, and the knot is fibred if and only if the commutator subgroup is finitely generated, in which case the commutator subgroup is free of even rank. Poincaré duality and the condition χ(M(K)) = 0 together impose subtle constraints on 2-knot groups which do not apply in higher dimensions. In particular, if the commutator subgroup π of a 2-knot group π is finitely generated then the virtual cohomological dimension of π is 1, 2 or 4. In this note we shall show that (modulo several plausible conjectures) a 2-knot group π is the group of a fibred 2-knot if and only if π is finitely generated, and if moreover π is torsion-free every 2-knot with group π is s-concordant to a fibred 2-knot. A simple satellite construction gives an example of a 2-knot whose group π is not virtually torsion-free (and so π is not finitely generated). Although our main interest is in the case of 2-knots, we give examples of fibred n-knots with groups of cohomological dimension d, for every n ≥ 4 and d ≥ 1. (No purely algebraic characterization of the groups of fibred n-knots is yet known for any n.) It is not clear whether there are fibred 3-knots with such groups. In the final section we consider other possible fibrations of 2-knot manifolds. This work was prompted by reading [8], where it is shown that there is a high-dimensional knot group which contains copies of every finitely presentable group, and it is suggested that there should be a similar 2-knot group. Our results do not address the questions raised at the end of [8] beyond the observations that no examples supporting the suggestions made there can be the groups of fibred 2-knots, and very likely no such examples have finitely generated commutator subgroup. 1991 Mathematics Subject Classification. 57Q45.