Some conditions for finite presentation of nilpotent-by-abelian groups

Abstract

Bieri and Strebel [2] have established a striking criterion that determines whether a metabelian group is finitely presented. This criterion is of a sort that implies that all homomorphic images of a finitely presented metabelian group are finitely presented. Another of their results is that the class of nilpotent-of-class-two-by-abelian groups has the property that all homomorphic images of finitely presented groups in this class are finitely presented. However, they point out that an example due to Abels [1] shows that the class of nilpotent-of-class-three-by-abelian groups does not have this property. One could not therefore expect that a criterion of the same type as that of Bieri and Strebel would distinguish the finitely presented groups in general classes of nilpotent-by-abelian groups. One might hope, however, to find such a criterion for nilpotent-of-class-two-by-abelian groups. The aim of this paper is to make some steps in this direction, by finding a sufficient condition for such groups to be finitely presented.