Geometry of hyperbolic groups and of their actions on Banach spaces

该项目将集中在Gromov意义下的双曲群的内在几何（特别是它的边界几何，它的共形维数等）和同一双曲群在某种类型和cotype的Banach空间上的作用几何（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的设置是随机组，无论是在三角形和密度模型。研究的背景，包括潜在的影响这些类型的问题最近引起了很多兴趣，特别是因为它们以多种方式连接到随机图和扩展器，这是一个在组合学和理论计算机科学中都是主流的主题。目的和目标这个项目的目的是通过定理和相关的例子来阐明，双曲群边界的共形维数与这类群可以作用的Banach空间的几何之间的联系。它还将旨在阐明扩展随机图可以具有哪些强性质，研究方法的新奇所涉及的方法混合了分析概念和方法的离散化，并使用概率来推断具有特殊性质的图和组的存在性。与EPSRC的策略和研究领域保持一致该项目福尔斯属于EPSRC“几何与拓扑学”研究领域，也与“数学分析”研究领域处于边缘。

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.