Geometry of hyperbolic groups and of their actions on Banach spaces

双曲群的几何及其在巴纳赫空间上的作用

• 批准号: 2099922
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2018
• 资助国家: 英国
• 起止时间: 2018 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2099922
• 关键词: Geometry; hyperbolic; groups; their; actions

Summary of the projectThis project will focus on the possible connection between the intrinsic geometry of a hyperbolic group in the sense of Gromov (in particular the geometry of its boundary, its conformal dimension, and so forth) and the geometry of the actions of the same hyperbolic group on Banach spaces of a certain type and cotype (in particular its actions on L^p spaces). One specific setting in which these questions will be looked into is that of random groups, both in the triangular and in the density model.Context of the research including potential impactThese types of questions have aroused a lot of interest recently, especially since they connect in more than one way to random graphs and expanders, a topic that is mainstream nowadays both in combinatorics and in theoretical computer science.Aims and objectivesThe aim of this project will be to clarify, through theorems and relevant examples, the conjectured connection between the conformal dimension of the boundaries of hyperbolic groups and the geometry of the Banach spaces on which such groups can act. It will also aim to clarify what strong properties of expansion random graphs can have, and to deduce corresponding statements about random groups.Novelty of the research methodologyThe methodology involved mixes discretisations of analytical concepts and methods and the use of probability to deduce the existence of graphs and groups with special properties.Alignment to EPSRC's strategies and research areasThis project falls within the EPSRC `Geometry and Topology' research area, and is also at the borderline with the `Mathematical Analysis' research area.

Project This项目的摘要将重点介绍Gromov的双曲线群体的内在几何形状之间的可能联系（尤其是其边界的几何形状，其保形维度等）与同一双曲线群体在某种类型和Cotype（尤其是对L^p space的动作）上相同双曲线群体的作用的几何形状。这些问题将要研究的一个具体环境是随机组，无论是在三角形还是密度模型中。定理和相关示例，双曲基团边界边界的共形维与此类组可以作用的Banach空间的几何形状之间的猜想联系。它还将旨在阐明扩展随机图的强大特性可以具有什么，并推断出有关随机组的相应陈述。研究方法的novelty涉及方法论，涉及分析概念和方法的离散性以及使用概率的使用来推导图形和组的存在与特性的特性和特殊属性。并且也处于“数学分析”研究领域的边界。