Coarse geometry and groups

Abstract

The central idea of coarse geometry is to focus on the properties of metric spaces which survive under deformations that change distances in a controlled way. These large scale properties, although too coarse to determine what happens locally, are nevertheless often able to capture the most important information about the structure of a space or a group. The relevant notions from coarse geometry and group theory are described in the beginning of this thesis.

An overview of the cohomological characterisation of property A of Brodzki, Niblo and Wright is given, together with a proof that the cohomology theories used to detect property A are coarse invariants. The cohomological characterisation is used alongside a symmetrisation result for functions defining property A to give a new direct, more geometric proof that expanders do not have property A, making the connection between the two properties explicit.
This is based on the observation that both the expander condition and property A can be expressed in terms of a coboundary operator which measures the size of the (co)boundary of a set of vertices.

The rest of the thesis is devoted to the study of box spaces, including a description of the connections between analytic properties of groups and coarse geometric properties of box spaces. The construction of Arzhantseva, Guentner and Spakula of a box space of a finitely generated free group which coarsely embeds into Hilbert is the first example of a bounded geometry metric space which coarsely embeds into Hilbert space but does not have property A. This example is generalised here to box spaces of a large class of groups via a stability result for box spaces.

More information

Identifiers

ORCID for Graham Niblo: orcid.org/0000-0003-0648-7027

ORCID for Jacek Brodzki: orcid.org/0000-0002-4524-1081

Catalogue record

Export record