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应用于任何出现对称性的地方。在拓扑学中，“空间的基本群”起着重要的作用。在拓扑学中，抽象呈现群作为空间的基本群出现，这些空间可以通过将多个多边形粘合在一起来构建。几乎双呈现群是具有双呈现群的许多代数性质的群。在20世纪90年代，经过30多年的研究之后，Besteland和布雷迪发现了第一个几乎是双呈现的群体而不是双呈现的例子。Bestvina-Brady的例子和所有其他发现，因为他们使用的技术“莫尔斯理论的立方复合”在其建设。这个项目的目的是寻找不涉及莫尔斯理论的几乎完全呈现群的新构造，以更清楚地了解这些群的性质。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意义大得多的“莫尔斯理论”。这项工作主要是在几何和拓扑，但有很强的联系，代数和组合。

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.