New constructions for almost finitely presented groups

几乎有限呈现的群体的新结构

• 批准号: 1949310
• 金额: --
• 依托单位: University of Southampton
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2017
• 资助国家: 英国
• 起止时间: 2017 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-1949310
• 关键词: New; constructions; almost; finitely; presented

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.

群是测量对称性的数学对象，所以关于群的结果可以应用于任何对称出现的地方。在拓扑学中，“空间的基本群”起着重要的作用。有限表示群作为空间的基本群出现在拓扑学中，这些空间可以通过将有限多个多边形粘合在一起而形成。几乎有限表示群是具有有限表示群的许多代数性质的群。在经过30多年的研究后，Bestvina和Brady在20世纪90年代发现了第一个几乎有限呈现的群体，而不是有限呈现的群体。Bestvina-Brady的例子和他们之后发现的所有其他例子都使用了“立方体复合体的莫尔斯理论”来构建它们。这个项目的目的是寻找不涉及莫尔斯理论的几乎有限呈现群的新结构，以揭示这些群的性质。一种可能的方法是使用被称为“小抵消理论”的较老的组合技术来取代更几何化的“莫尔斯理论”。这项工作主要涉及几何和拓扑，但与代数和组合学有很强的联系。