Almost finitely presented soluble groups

Abstract

1.1 Finitely presented soluble groups have been investigated by several authors. Roughly speaking, their results deal with two aspects: with the subgroup structure of finitely presented soluble groups [2], [4], [7], [18], and with soluble varieties whose non-cyclic relatively free group are infinitely related [16], [3]. Here we attack the problem of recognizing which finitely generated soluble groups are finitely related in a somewhat more systematic way: We show that all finitely presented soluble groups have a certain structural property which is inherited by homomorphic images (whether this holds for the property of being finitely presented itself is an old problem of P. Hall's and is still open). 1.2 The methods of [2] and [4] made it clear that even in the soluble case the HNN-construction is an important tool for obtaining finitely presented groups. Recall that every group B containing a pair of isomorphic subgroups 0 : S---~ T is embedded in the HNN-group