Non-positive curvature, geometry, and geometric group theory

Lead Research Organisation: University of Oxford
Department Name: Mathematical Institute

Abstract

Geometric group theory grew enormously from 1990 onwards to become a major field of modern mathematics with deep interconnections and rich interfaces with many other branches. It has provided a remarkable number of new tools that have led to the resolution of longstanding central problems in those fields, a notable example being the way in which the powerful theory of non-positively curved cube complexes (or high dimensions) was used to solve the remaining major questions about the geometry and topology of 3-dimensional manifolds. A further important set of applications revolves around the deep insights that Geometric Group Theory has provided into the structure of mapping class groups of surfaces and automorphism groups of free groups, objects that are of importance in a wide variety of mathematical contexts, and objects whose rich structures retain many mysteries, the resolution of which would have wide impact in geometry, number theory, and mathematical physics. Manifestations of non-positive curvature (recast far beyond its original realm of differential geometry) play a central role in many of these topics.
The aim of this research project is to attack a range of problems in the mainstream of geometric group theory that involve aspects of non-positive curvature, harness some of the great breakthroughs of recent years, and have the potential to unlock significant applications in adjoining fields of mathematics. Success will require not only a rigorous understanding of a number of challenging topics spanning algebra, topology and geometry: it will also require great novelty in interpreting and extending the latest advances concerning surfaces and their moduli, Outer space, and cubulation techniques.

Besides advancing the core of mathematics, the training of the student in this central area of modern mathematics will enhance the UK's human capital in the area, and its position at the forefront of fundamental research in the mathematical sciences - a strength whose economic and societal importance has been well documented in the context of the government's industrial strategy and planning for the post-Brexit era. This falls within EPSRC Mathematical Sciences Geometry and Topology research area , but also has the potential to contribute to both the digital economy agenda, and the cybersecurity agenda.

Publications

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S Shepherd (2019) Two generalisations of Leighton's Theorem in arXiv

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S Shepherd (2019) Agol's theorem on hyperbolic cubulations in arXiv

Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509711/1 01/10/2016 30/09/2021
1941997 Studentship EP/N509711/1 01/10/2017 31/03/2021 Sam Shepherd
 
Description My first paper was a set of detailed notes explaining the proof of a major theorem of Agol about hyperbolic cubulations. Agol's Theorem says that hyperbolic cubulated groups are virtually special; this is an important result because many groups are known to be hyperbolic and cubulated (such as 3-manifold and small cancellation groups), and virtually special groups are known to satisfy a number of strong properties (such as linearity and separability properties). I retained the underlying ideas of Agol's proof, but substantially changed or added to many parts of the argument to give a more transparent and detailed account.

My second paper provided two generalisations of Leighton's Graph Covering Theorem, and the first of these answered a question from a paper of Neumann in 2012. The first generalisation was in fact independently proven by Giles Gardam and Daniel Woodhouse at the same time, and their proof (which uses different techniques to mine) was added as an appendix to my paper. While Leighton's Theorem allows for the construction of common finite covers of graphs, our theorems can be used to construct common finite covers of a wide range of spaces that possess a certain kind of graph-like structure. Covers are an essential tool in geometric group theory, and in topology more generally, and finite covers are of particular interest to those who study groups up to commensurability. Since writing this paper, I have been collaborating with Daniel Woodhouse, and our current project uses some of the key ideas from our work on Leighton's Theorem.
Exploitation Route I expect that my paper about Agol's Theorem will be a useful resource to other researchers in the area, as it will help them to understand the ideas from this proof, and possibly build on them to prove similar results. I already know a number of people interested in my paper, and I was invited to give a talk about it in Cambridge last year.

The paper generalising Leighton's Theorem has already been used in recent work of Stark and Woodhouse about action rigidity. There are several earlier papers in the literature that also use variants of Leighton's Theorem to prove some kind of rigidity result, so I think it's likely that future papers about rigidity will make use of our results.
Sectors Other

 
Description Two generalisations of Leighton's Theorem: with Giles Gardam and Daniel Woodhouse 
Organisation Technion - Israel Institute of Technology
Country Israel 
Sector Academic/University 
PI Contribution I wrote the main body of the paper.
Collaborator Contribution Giles Gardam and Daniel Woodhouse wrote the appendix to the paper, which included an alternative proof of one of the main theorems.
Impact The paper 'Two generalisations of Leighton's Theorem'. It is not a multi-disciplinary collaboration.
Start Year 2019
 
Description Two generalisations of Leighton's Theorem: with Giles Gardam and Daniel Woodhouse 
Organisation University of M√ľnster
Country Germany 
Sector Academic/University 
PI Contribution I wrote the main body of the paper.
Collaborator Contribution Giles Gardam and Daniel Woodhouse wrote the appendix to the paper, which included an alternative proof of one of the main theorems.
Impact The paper 'Two generalisations of Leighton's Theorem'. It is not a multi-disciplinary collaboration.
Start Year 2019