New constructions for almost finitely presented groups

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Groups are mathematical objects that measure symmetry, and so resultsabout groups can have applications wherever symmetry arises. In topologythe `fundamental group of a space' plays an important role. Finitely presented groups arise in topology as the fundamental groups of the spaces that can be built by gluing together finitely many polygons. Almost finitely presented groups are groups that share many of the algebraic properties of finitely presented groups. The first examples of almost finitely presented groups that are not finitely presented were found by Bestvina and Brady in the 1990's, after a search that lastedmore than 30 years. The Bestvina-Brady examples and all others found sincetheirs have used the technique `Morse theory for cubical complexes' in their construction. The aim of this project is to search for new constructions of almost finitely presented groups that do not involve Morse theory, to shed more light on the properties of these groups. One potential route is to use the older combinatorial technique known as `small cancellation theory' to replace the far more geometric `Morse theory'. This work lies mainly within geometry and topology, but has strong links with algebra and combinatorics.