Cyclic extensions of parafree groups

Abstract:

Let $1 \to F \to G \to T \to 1$ be a short exact sequence where F is parafree and T is infinite cyclic. We examine some properties of G when $F/F’$ is a free ZT-module. Here $F’$ is the commutator subgroup of F and ZT is the integral group ring of T. In particular, we show G is parafree and ${\gamma _n}F/{\gamma _{n + 1}}F$ is a free ZT-module for every $n \geqslant 1$ (where ${\gamma _n}F$ is the nth term of the lower central series of F).