Interactions between geometric group theory and topology

几何群论与拓扑学之间的相互作用

• 批准号: 2625336
• 金额: --
• 依托单位: University of Manchester
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2625336
• 关键词: Interactions; between; geometric; group; theory; Interactions; between; geometric; group; theory

In geometric group theory, one studies a given group (an algebraic object) as a collection of symmetries of some geometric space. This point of view has been applied successfully to groups from a wide range of topics in mathematics, including random groups, various groups from low-dimensional topology, and more recently the Cremona group.In this project, the groups that we are interested in are the groups of homeomorphisms of surfaces. Surfaces are 2-dimensional objects, and include the surface of a ball, the surface of a ring doughnut, and the surface of a block of wood with holes drilled through it. A homeomorphism of a surface is a continuous bending or stretching of the surface, which can be reversed. Homeomorphisms appear naturally, for instance, consider the surface of a fluid under mixing.The collection of homeomorphisms of a surface forms a group. Surprisingly very little is known about this group, despite its presence in mathematics for more than 100 years. There are in general few tools to study these groups. However Bowden, Hensel, and Webb recently showed that, provided that the "number of holes" in the surface is at least 1, these groups can be studied using techniques from geometric group theory. The exact tool used was the construction of an (uncountably infinite) graph on which the homeomorphism group acts by symmetries, and such that the graph has nice geometry (namely Gromov hyperbolic geometry). This enables algebraic theorems to be proved. Unfortunately this new theory does not carry through in its strongest form for the homeomorphism group of the 2-dimensional sphere. However if we restrict attention to the area-preserving homeomorphisms of the 2-dimensional sphere, then it might be the case that similar tools and constructions will work once again. The goal of the project is to either rule out the existence of such a construction, or prove and utilize it to study the group.This project lies in the EPSRC research areas of "Algebra" and "Geometry and Topology".

在几何群体理论中，一个人研究一个给定的组（代数对象）是某些几何空间的对称性的集合。这种观点已成功地应用于来自数学广泛主题的小组，包括随机组，低维拓扑的各个组以及最近的Cremona群体。在这个项目中，我们感兴趣的组是表面的同态同态。表面是二维物体，包括球的表面，环甜甜圈的表面以及一块木头的表面，并钻了孔。表面的同态性是表面的连续弯曲或拉伸，可以逆转。例如，同构自然而然地考虑在混合下的液体表面。表面的同态形态的集合形成了一组。令人惊讶的是，尽管该群体在数学中存在超过100年，但知之甚少。通常，很少有研究这些组的工具。然而，鲍登，汉斯尔和韦伯最近表明，前提是表面上的“孔数”至少为1，可以使用几何组理论的技术来研究这些组。所使用的确切工具是（无限）图的构造，同态组通过对称性作用，并且该图具有不错的几何形状（即Gromov双曲线几何形状）。这使代数定理得到证明。不幸的是，这种新理论并没有以最强的形式贯穿于二维球体的同态群体。但是，如果我们限制了二维球体保护区域的同态同态的关注，那么可能是类似的工具和结构再次起作用的情况。该项目的目的是排除这种结构的存在，要么证明并利用它来研究该组。该项目在于“代数”和“几何和拓扑”的EPSRC研究领域。