Geometry of hyperbolic groups and of their actions on Banach spaces

Summary of the project

This 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 impact

These 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 objectives

The 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 methodology

The 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 areas

This project falls within the EPSRC `Geometry and Topology' research area, and is also at the borderline with the `Mathematical Analysis' research area.