New constructions for almost finitely presented groups

群是测量对称性的数学对象，因此关于群的结果可以在任何出现对称性的地方得到应用。在拓扑学中，‘空间的基本群’起着重要的作用。有限表示群在拓扑学中作为空间的基本群出现，这些空间可以通过将有限多个多边形粘合在一起来建立。几乎有限表示群是共享有限表示群的许多代数性质的群。经过30多年的探索，贝斯特维纳和布雷迪在20世纪90年代的S中发现了第一批几乎有限呈现的不是有限呈现的群。Bestvina-Brady的例子和自他们以来发现的所有其他例子都在他们的构造中使用了‘立方复合体的莫尔斯理论’的技术。这个项目的目的是寻找不涉及Morse理论的几乎有限表示群的新结构，以更好地揭示这些群的性质。一种可能的途径是使用被称为“小抵消理论”的较老的组合技术来取代几何程度高得多的“莫尔斯理论”。这项工作主要涉及几何学和拓扑学，但与代数和组合学有很强的联系。

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.