Abstract. If G = (x1, …, xn \ rm) is a one-relator group for some large integer m, then it is well known that G has many nice properties, and these are often easier to prove than in the torsion-free case (where the relator is not a proper power). To a large extent, this phenomenon also occurs for a one-relator product G = (A * B)/N(rm) of arbitrary groups A, B (where N(·) means normal closure). In this article we survey some recent results about such groups, describe the geometric methods used to prove these results, and discuss what happens for lower values of m. Specifically, we give counterexamples to a conjecture made in [36].

Introduction

Given a set X = {x1, x2, x3,…} and a word r on X ∪ X−1 the group G given by the presentation (x1, x2, x3,… | r) is said to be a one-relator group. An extensive and successful theory of one-relator groups has been established, based largely on the work of W. Magnus, and generalising the theory of free groups (see [44], Chapter II). The theory of one-relator groups provides a basic model and an extensive stock of techniques for further generalization of free groups. With this aim in mind, one possibility is to consider presentations with more than one relator. This idea has been been successfully pursued by I.L. Anshel [1] for the case of two-relator groups (see also [40] for a particular class of two-relator groups). Bogley [4] has also extended a number of one-relator group theorems to certain groups with arbitrarily many relations.