Free Group Automorphisms and Knotted Tori in S4

Abstract

Let ϕ be an automorphism of a finitely generated free group F, w and element of F, and H the subgroup of F generated by the orbit {ϕn (w), |n ∈ Z}. We describe sufficient conditions ensuring that H is non-finitely generated. Using this we give a simple construction of tori T embedded in S4 in such a way that the commutator subgroup of π1(S4 - T) is finitely generated but not finitely presented. Such tori have no minimal Seifert manifolds. An example of an embedded torus with the latter property was given recently by T. Maeda using different methods.