Residual and algorithmic properties of generalized Bestvina-Brady groups
Lead Research Organisation:
University of Southampton
Department Name: Sch of Mathematical Sciences
Abstract
Groups are mathematical objects that measure symmetry, and so results
about groups can have applications wherever symmetry arises. Groups
can be studied algebraically or via the geometry of spaces with
symmetries. Some groups can be well approximated by either by
permutations of a finite set or by matrices; such groups are called
`residually finite' and `linear' respectively. These groups are
algorithmically tractable. In the 1990's Bestvina and Brady introduced
a family of groups with surprising geometric properties, and used these
groups to resolve a number of open problems. Bestvina-Brady groups are
always linear. Recently Leary generalized these groups to a far larger
family whose members cannot all be linear. The aim of this project
is to characterize which members of the new family are either linear or
residually finite. Although this question is phrased in algebraic terms,
many of the techniques used are geometric; in particular the method of
`special cube complexes' as used by Agol and Wise in their celebrated
recent work on 3-manifolds is expected to play a role.
about groups can have applications wherever symmetry arises. Groups
can be studied algebraically or via the geometry of spaces with
symmetries. Some groups can be well approximated by either by
permutations of a finite set or by matrices; such groups are called
`residually finite' and `linear' respectively. These groups are
algorithmically tractable. In the 1990's Bestvina and Brady introduced
a family of groups with surprising geometric properties, and used these
groups to resolve a number of open problems. Bestvina-Brady groups are
always linear. Recently Leary generalized these groups to a far larger
family whose members cannot all be linear. The aim of this project
is to characterize which members of the new family are either linear or
residually finite. Although this question is phrased in algebraic terms,
many of the techniques used are geometric; in particular the method of
`special cube complexes' as used by Agol and Wise in their celebrated
recent work on 3-manifolds is expected to play a role.
Organisations
Publications
Vankov V
(2022)
Virtually special non-finitely presented groups via linear characters
in Geometriae Dedicata
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509747/1 | 01/10/2016 | 30/09/2021 | |||
| 1949146 | Studentship | EP/N509747/1 | 01/10/2017 | 31/03/2021 | Vladimir Vankov |
| Description | A new method for tackling a popular problem (specifically, the problem of determining special cube complexes) in currently active research fields (specifically, geometric group theory and low-dimensional topology) has been developed, by bringing together ideas from related research fields (specifically, representation theory). New avenues for research have been opened up (specifically, for studying infinite groups through their finite quotients). An open question regarding group actions has been answered. Properties of large families of infinite groups have been determined. |
| Exploitation Route | The outcomes are purely academic. The research field concerning the outcomes is rapidly progressing, so collaborations and further application are expected. Furthermore, the newly developed method uses tools from related fields which would be of interest to researchers outside the specific setting of the original problem. |
| Sectors | Other |
| URL | https://arxiv.org/abs/2202.08169 |
| Description | Hosting the Postgraduate Group Theory Conference |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Postgraduate students |
| Results and Impact | I successfully bid for hosting the annual conference "PGTC". As such, the event was held at the University of Southampton. Organising of the event included securing external funding. Many people from across the world joined in via Zoom. We had invited keynote speakers as well as many contributed talks by participants. 150 people registered, with 40 contributed talks. |
| Year(s) Of Engagement Activity | 2021 |
| URL | https://sites.google.com/view/pgtc2020 |
| Description | Presentation for the Cambridge Geometric Group Theory group |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | National |
| Primary Audience | Other audiences |
| Results and Impact | I was invited to speak about a new method I am developing. The Geometric Group Theory study group at the University of Cambridge was covering a related topic at the time and invited me as speaker to learn about the new approaches to the topic which are involved in my research. Various researchers were asking me about when the relevant paper would be published, which created anticipation and excitement for the outputs of my research. |
| Year(s) Of Engagement Activity | 2019 |
| Description | Talk at the Bristol Junior Geometry Seminar (BRIJGES) |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | National |
| Primary Audience | Postgraduate students |
| Results and Impact | I spoke about the fundamentals of the mathematics underlying my research. This is important as it helps to cement my standing as an expert in the field. |
| Year(s) Of Engagement Activity | 2019 |
| URL | https://sites.google.com/view/brijges/home |
| Description | Talk at the Cambridge Junior Algebra Seminar |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | National |
| Primary Audience | Postgraduate students |
| Results and Impact | I was invited to speak about my research, because my research covers an interesting overlap of several active research fields that researchers in Cambridge were interested in. I was also developing a new method at the time. The talk was a success, as I was invited to speak about my new method later for a study group, including receiving funding for travel to allow this to happen. |
| Year(s) Of Engagement Activity | 2019 |
| Description | Talk at the Postgraduate Group Theory Conference 2019 in Birmingham |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Postgraduate students |
| Results and Impact | I gave a talk about my research, which is primarily focused on infinite groups. The majority of the audience were PhD students working with finite groups. The aim of the talk was to bridge the gap between our two fields. This was achieved, as this sparked fruitful discussions afterwards about close relationships between the two fields. |
| Year(s) Of Engagement Activity | 2019 |
| URL | http://web.mat.bham.ac.uk/pgtc19/ |