Splittings, tiles, and planarity: a trio of problems in geometric group theory

Thesis

This thesis aims to exposit three results in the field of geometric group theory (GGT). Each of these results has their own distinct flavour, and through this we intend to showcase some of the key themes of research within GGT.
The first of these problems (‘Splittings’) asks for an algorithm to detect whether a given group splits as an amalgamated free product or HNN extension in a certain way. More precisely, we will study the question of whether, given a one-ended hyperbolic group G and generators of a quasi-convex subgroup H, one can effectively decide whether G splits over a subgroup commensurable with H. The answer is positive if we assume additionally that H is residually finite, though a small technicality seems to obstruct the general case. We also present an algorithm to compute the number e(G, H) of relative ends, as well as partial results towards computing the number of filtered ends.
The second problem (‘Tiles’) relates to a long-standing question in group theory asked independently by C. Chou and B. Weiss, which is whether every group is monotileable. We present progress on this question by proving that every acylindrically hyperbolic group is monotileable. Our proof expands on the techniques of a recent paper of A. Akhmedov, who showed that every hyperbolic group is monotileable. This chapter is based on joint work with L. Mineh.
The third and final problem we consider (‘Planarity’) concerns a coarse characterisation of virtually planar groups. That is, those finitely generated groups containing a finite-index subgroup which admits a planar Cayley graph. We show that if a finitely generated group is quasi-isometric to a planar graph, then it is virtually planar. The main technical achievement of this chapter is showing that such a group is accessible, in the sense of Wall and Dunwoody, as we do not know, a priori, that our group is finitely presented. This is achieved through a careful study of quasi-actions on planar graphs. In particular, we ‘split’ our group into more and more ‘highly connected’ pieces, until this quasi-action becomes so controlled that we obtain a true action on a suitable 2-complex.

Authors: MacManus, J

• DOI: 10.5287/ora-qoqomxga1
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: topology; group theory; geometric group theory; algebra; geometry
• Subjects: Mathematics