Millard, Benjamin James; (2021) Stackings and One-Relator Products of Groups. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

The theory of one-relator groups (groups admitting a presentation with a single relator) has a selection of interesting open questions. For example, it is unknown exactly which torsion-free one-relator groups are coherent, or which are hyperbolic. One way in which these questions are currently being studied is by considering immersions of 2-complexes into a group’s presentation complex. The specific properties of interest are called non-positive immersions and negative immersions, which put restrictions on the Euler characteristic of the immersing space. In this thesis, we look more closely at these properties, and see how they have been used to study one-relator groups. We also consider one-relator products of groups, a generalisa- tion of one-relator groups where a defining relator is taken over a free product of groups rather than just a free group. We prove that one-relator products admit a stacking, which is a geometric object containing information about the relationship between the defining relator and the underlying free product of groups. We go on to use these stackings to prove that torsion-free one-relator products have the non- positive immersions property. This is a result that has also recently been proved by James Howie and Hamish Short. We discuss how our proof differs and how by using stackings we can find improvements for some of their results. Finally, we discuss the negative immersions property, its conjectural link to hyperbolicity, and how stackings may allow progress in the classification of which one-relator products have negative immersions.

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