Geometry of hyperbolic groups and of their actions on Banach spaces

本项目将重点研究Gromov意义上的双曲群的固有几何（特别是它的边界几何、共形维数等）与同一双曲群在某型和协型的Banach空间上的作用的几何（特别是它在L^p空间上的作用）之间的可能联系。这些问题将被研究的一个特定环境是随机群体，包括三角形模型和密度模型。这些类型的问题最近引起了很多人的兴趣，特别是因为它们以不止一种方式与随机图和扩展器联系在一起，这是当今组合学和理论计算机科学的主流主题。目的和目标这个项目的目的是通过定理和相关的例子来澄清双曲群边界的共形维数和这些群可以作用的巴拿赫空间的几何之间的猜想联系。它还将旨在澄清展开随机图可能具有的强性质，并推导出关于随机群的相应陈述。研究方法的新颖性该方法混合了分析概念和方法的离散性以及使用概率来推断具有特殊性质的图和群的存在性。与EPSRC的战略和研究领域保持一致该项目属于EPSRC的“几何和拓扑”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.