One-Relator Groups: An Overview

In 1985, at Groups St Andrews, Gilbert Baumslag gave a short course on one-relator groups which provided a look at the subject up to that point. In this paper we partially update the massive amount of work done over the past three decades. For the most part we concentrate on areas and results to which the authors have made contributions. We look at the important connections with surface groups and elementary theory, and describe the surface group conjecture and the Gromov conjecture on surface subgroups. We look at the solution by Wise of Baumslag’s residual finiteness conjecture and discuss a new Baumslag conjecture on virtually free-by-cyclic groups. We examine various amalgam decompositions of one-relator groups and the Baumslag-Shalen conjectures. We then look at a series of open problems in one-relator group theory and their status. Finally we introduce a concept called plainarity based on the Magnus breakdown of a one-relator group which might provide a systematic approach to the solution of problems in one-relator groups.