Abstract

Within the class of one-relator groups, those with torsion are better understood: they are hyperbolic, their Magnus subgroups are quasi-convex and so they (virtually) have quasi-convex hierarchies. However, many torsion-free two-generator one-relator groups exhibit pathological behaviours. Recently, Louder and Wilton have shown that one-relator groups with negative immersions do not contain two-generator one-relator subgroups, leading them to conjecture that such groups are hyperbolic. In this talk, I will show how to refine the classical Magnus—Moldavanskii hierarchy for a one-relator group. I will show that a one-relator hierarchy without Baumslag—Solitar subgroups is a hyperbolic quasi-convex hierarchy if it satisfies an additional technical hypothesis. I will then relate this with the conjecture of Louder and Wilton and show how it can be converted to a question about free groups.