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.
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.
People |
ORCID iD |
| Martin Robert Bridson (Primary Supervisor) | |
| Sam Shepherd (Student) |
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 | 28/04/2021 | 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. My third paper contained a quasi-isometric rigidity result for graphs of virtually free groups with two-ended edge groups - in particular this applies to 'generic' cyclic HNN extensions of free groups. This was joint work with Daniel Woodhouse, and part of the argument used Leighton-style techniques similar to my second paper. There are many quasi-isometric rigidity results in the literature, for example for free groups, abelian groups and surface groups. My fourth paper explored possible extensions to Leighton's Theorem. It was joint work with my supervisor Martin Bridson. We focused on the case where two finite graphs are covered by a quasitree, and investigated whether there is a common finite cover of the finite graphs which is itself covered by the quasitree. Using a theorem from my second paper, we proved a positive answer if the original covering maps are regular, plus we provided counter-examples to some other cases. |
| 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. In fact another author built on my proof to obtain a generalisation of Agol's Theorem to the relatively hyperbolic setting. The paper generalising Leighton's Theorem has already been used in work of Stark and Woodhouse about action rigidity and in my own paper with Bridson. 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. My paper about quasi-isometric rigidity has probably been the most widely disseminated, because both Woodhouse and I have given multiple seminar talks about it at a variety of universities (including several in North America). The techniques used in the paper may inspire others to prove further rigidity results. My paper about Leighton's Theorem and quasitrees provides a further example of how the theorem from my earlier Leighton's Theorem paper can be applied, which may encourage others to also use it. The counter-examples we construct in other cases contain a different set of ideas, which may be of interest to other researchers. |
| Sectors | Other |
| Description | Leighton's Theorem: extensions, limitations, and quasitrees: with Martin Bridson |
| Organisation | University of Oxford |
| Country | United Kingdom |
| Sector | Academic/University |
| PI Contribution | The whole paper was joint work |
| Collaborator Contribution | The whole paper was joint work. |
| Impact | The paper 'Leighton's Theorem: extensions, limitations, and quasitrees'. It is not a multi-disciplinary collaboration. |
| Start Year | 2019 |
| Description | Quasi-isometric rigidity for graphs of virtually free groups with two-ended edge groups: with Daniel Woodhouse |
| Organisation | University of Oxford |
| Country | United Kingdom |
| Sector | Academic/University |
| PI Contribution | The whole paper was joint work. |
| Collaborator Contribution | The whole paper was joint work. |
| Impact | The paper 'Quasi-isometric rigidity for graphs of virtually free groups with two-ended edge groups'. It is not a multi-disciplinary collaboration. |
| Start Year | 2019 |
| 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 Münster |
| 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 |