Geometry of hyperbolic groups and of their actions on Banach spaces

本课题将主要研究Gromov意义下的双曲群的内在几何(特别是其边界的几何、共形维度等)与同一双曲群在某种类型和余型的Banach空间上的作用(特别是它在L^p空间上的作用)之间的可能联系。这类问题最近引起了人们的极大兴趣，特别是因为它们以一种以上的方式与随机图和展开器联系在一起，这是当今组合学和理论计算机科学中的一个主流话题。本项目的目的和目标是通过定理和相关例子阐明双曲群的边界的共形维度和这些群可以作用于的Banach空间的几何之间的猜想联系。它还将旨在阐明扩展随机图可以具有的强性质，并推导出关于随机群的相应声明。研究方法的新颖性所涉及的方法论混合了分析概念和方法的离散化以及使用概率来推导具有特殊性质的图和群的存在。与EPSRC的策略和研究领域保持一致本项目属于EPSRC的几何和拓扑学研究领域，也处于数学分析研究领域的边缘。

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.