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.

Publications

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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 28/09/2017 30/09/2020 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).
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/2001.11868
 
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/