The geometry of groups and homomorphisms
Lead Research Organisation:
University of Cambridge
Department Name: Pure Maths and Mathematical Statistics
Abstract
This project is about using geometry to understand relationships between groups. A 'group' is the technical mathematical term for a collection of symmetries, and relationships between groups are the 'homomorphisms' of the project title. We will try to understand fundamental, unanswered questions like:
- 'When can we embed one group of symmetries in another?';
- 'What do the finite groups that a group relates to tell us about it?';
and
- 'Can we classify all the relationships between a group and 3-dimensional spaces?'
Answers to these questions will deepen our understanding of the abstract geometric spaces on which these groups of symmetries act. Although these questions come from within pure mathematics, the tools that we will develop to answer them can also potentially be applied to abstract spaces that arise in other, more applicable, disciplines, such as biology or data analysis.
- 'When can we embed one group of symmetries in another?';
- 'What do the finite groups that a group relates to tell us about it?';
and
- 'Can we classify all the relationships between a group and 3-dimensional spaces?'
Answers to these questions will deepen our understanding of the abstract geometric spaces on which these groups of symmetries act. Although these questions come from within pure mathematics, the tools that we will develop to answer them can also potentially be applied to abstract spaces that arise in other, more applicable, disciplines, such as biology or data analysis.
Planned Impact
By tackling theoretical questions, this project develops the geometric-group-theory toolkit that will be available to apply to real-world problems. Although no specific real-world applications of the proposed research are currently known, there have already been several promising applications of the geometries studied by geometric group theorists (especially CAT(0) complexes and hyperbolic graphs) to real-world problems.
This project is motivated by questions that come from within pure mathematics. But one should not underestimate the potential benefits of the unforeseen consequences of theoretical research: the mental toolkit that Alan Turing developed in his research on abstract proof-theoretic questions proved useful both in code-breaking and the development of the computer. In fact, geometric group theory is already demonstrating its potential for real-world applications.
This project is motivated by questions that come from within pure mathematics. But one should not underestimate the potential benefits of the unforeseen consequences of theoretical research: the mental toolkit that Alan Turing developed in his research on abstract proof-theoretic questions proved useful both in code-breaking and the development of the computer. In fact, geometric group theory is already demonstrating its potential for real-world applications.
Organisations
People |
ORCID iD |
| Henry Wilton (Principal Investigator) |
Publications
Arzhantseva Goulnara N.
(2016)
Acylindrical hyperbolicity of cubical small-cancellation groups
in arXiv e-prints
Behrstock Jason
(2015)
Hierarchically hyperbolic spaces II: Combination theorems and the distance formula
in arXiv e-prints
Behrstock Jason
(2015)
Asymptotic dimension and small-cancellation for hierarchically hyperbolic spaces and groups
in arXiv e-prints
Bridson M
(2017)
Profinite rigidity and surface bundles over the circle
in Bulletin of the London Mathematical Society
Durham Matthew G.
(2016)
Boundaries and automorphisms of hierarchically hyperbolic spaces
in arXiv e-prints
Groves D
(2018)
The structure of limit groups over hyperbolic groups
in Israel Journal of Mathematics
Hagen M
(2019)
Panel collapse and its applications
in Groups, Geometry, and Dynamics
Hagen Mark F
(2016)
Cubulating mapping tori of polynomial growth free group automorphisms
in arXiv e-prints
Hagen Mark F
(2016)
On hierarchical hyperbolicity of cubical groups
in arXiv e-prints
Hanselman J
(2020)
L-spaces, taut foliations, and graph manifolds
in Compositio Mathematica
| Description | This project continued the PI's work on the symmetries of "negatively curved" spaces - hyperbolic groups, to use the technical term (see grant codes EP/I003843/1 and 2 for the earlier work). The first achievement can be seen as a contribution towards characterising the dimension of these groups. A theorem of Stallings showed that these groups are one-dimensional exactly when they are "free" - this means, they're the symmetries of a tiling of an infinite tree. On the other hand, most hyperbolic groups are of dimension greater than one. The most familiar such examples are two-dimensional "surface groups" - these arise as symmetries of tilings of the hyperbolic plane, an important mathematical object famously discovered by Bolyai and Lobachevksy during the nineteenth century. Gromov asked whether every hyperbolic group of dimension greater than two must have a subgroup that's a surface group - in other words, whether surface groups characterise the property of having dimension greater than one. Even special cases of Gromov's question are very difficult, and progress is a big step forward. Bestvina and Feighn explained how to construct hyperbolic groups by "gluing" free groups together. This construction gives a large class of hyperbolic groups of dimension greater than 1, but most of their obvious subgroups are free, so they are natural test-cases for Gromov's question. The main achievement of this project is to answer Gromov's question in this case: for every hyperbolic group obtained by gluing together free groups, either the group is free, or it contains a surface subgroup. Another achievement, in joint work of the PI's with Louder, was progress on studying "one-relator groups". These are the groups obtained from free groups by "quotienting", i.e. setting to zero, a single element. These groups have been studied for around a century, but still remain mysterious. Again, the challenge is to identify the sort of subgroups that they can contain. In particular, if they don't contain certain cousins of the symmetries of the square tiling of the Euclidean plane, then they are conjectured to be hyperbolic. Louder and the PI introduced the class of one-relator groups with "negative immersions", and proved various results about their subgroups that suggest lines of attack on this conjecture. In another paper, Louder and the PI proved a special case of a well known conjecture, which also concerns the subgroups of one-relator groups. Dr Hagen, also funded by this grant, continued his study of "hierarchically hyperbolic spaces". These are a far-reaching generalisation of the hyperbolic spaces already mentioned, and were introduced by Hagen, together with Behrstock and Sisto, to capture many natural classes of examples. |
| Exploitation Route | The work on surface subgroups opens up an important new direction in that area, and should be useful in answering Gromov's question in full generality. The work on one-relator groups will make possible the application of geometric techniques to their study, and bring together disparate themes in topology and group theory. Speculatively, it may have applications to the study of taut foliations and Heegaard Floer homology in 3-dimensional topology. Hagen's work on hierarchically hyperbolic spaces has been very influential in geometric group theory, and many researchers are now taking it further. |
| Sectors | Other |
| URL | https://arxiv.org/find/grp_math/1/au:+wilton_h*/0/1/0/all/0/1 |
| Description | An Enumerative Approach to Heegaard Floer |
| Amount | £244,455 (GBP) |
| Funding ID | EP/R02359X/1 |
| Organisation | Engineering and Physical Sciences Research Council (EPSRC) |
| Sector | Public |
| Country | United Kingdom |
| Start | 08/2018 |
| End | 08/2023 |
| Description | EPSRC Postdoctoral Fellowship |
| Amount | £244,455 (GBP) |
| Funding ID | EP/R02359X/1 |
| Organisation | Engineering and Physical Sciences Research Council (EPSRC) |
| Sector | Public |
| Country | United Kingdom |
| Start | 08/2018 |
| End | 08/2021 |
| Description | Physics Cafe (Aspen, CO, USA): Knots and Physics |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Public/other audiences |
| Results and Impact | As an offshoot of research activities coordinated by the Aspen Center for Physics in Aspen, CO, USA, the Aspen Center for Science hosts a weekly outreach event called the "Physics Cafe." It is a one-hour event, followed by a general-audience lecture by a celebrated physicist or mathematician. At the physics cafe itself, 2 researchers speak briefly about how they ended up in their current field of research, and then for the remaining 40 or so minutes, the audience ask the 2 researchers questions about their research or about physics and/or mathematics in general. On 7 March, 2018, I was one of the two researchers co-hosting the physics cafe. (My co-host was Piotr Kucharski.) We had a lively discussion, and afterwards, several audience members stayed on to ask further questions or request further information. One young secondary-school student even asked for my autograph. |
| Year(s) Of Engagement Activity | 2018 |
| URL | https://aspensciencecenter.org/programs/physics-cafes/ |