If P is a group of operators on a group A, then we denote by dp(A) the minimal number of generators of A as a P-group.

Suppose G is a group and F/N is a presentation of G. Then F acts on N by conjugation and induces an action of G on Nab. This ℤG-module is called the relation module of the presentation F/N.

Definition. A presentation F/N of a group G is said to have a relation gap if dG(Nab) is strictly less than dF(N).

It is an open problem whether there exists a presentation that has a relation gap (see Harlander [H1], [H2] and Baik, Pride [B-P]). Such a presentation would be interesting not only to group theorists. In [D] Dyer shows that a presentation with a relation gap could be used to settle an open question concerning complexes dominated by a 2-complex (see also Wall [W] and Ratcliffe [R]).

In [H1] and [H2] the author studies groups that have cyclic relation modules. Such groups are quotients G/P, with G a one-relator group, say presented by 〈X ∣ r〉, and P a perfect normal subgroup G. Now if P is not of the form 〈w〉F / 〈r〉F, where 〈w〉F denotes the normal closure of the element w of the free group F on X, then G/P has a presentation with a relation gap.