New constructions for almost finitely presented 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. In topology
the `fundamental group of a space' plays an important role. Finitely
presented groups arise in topology as the fundamental groups of the spaces
that can be built by gluing together finitely many polygons.
Almost finitely presented groups are groups that share
many of the algebraic properties of finitely presented groups. The first
examples of almost finitely presented groups that are not finitely presented
were found by Bestvina and Brady in the 1990's, after a search that lasted
more than 30 years. The Bestvina-Brady examples and all others found since
theirs have used the technique `Morse theory for cubical complexes'
in their construction. The aim of this project is to search for new
constructions of almost finitely presented groups that do not involve
Morse theory, to shed more light on the properties of these groups.
One potential route is to use the older combinatorial technique known as
`small cancellation theory' to replace the far more geometric `Morse theory'.
This work lies mainly within geometry and topology, but has strong links
with algebra and combinatorics.

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
1949310 Studentship EP/N509747/1 28/09/2017 30/09/2020 Thomas Brown
 
Description Finitely presented groups have been studied for a long time. Much is known about them and much can be said given such a finite presentation. However, there are only countably many finitely presented groups.

Groups of type FP, or 'almost finitely presented groups' posses many of the same properties as finitely presented groups. Except now we have uncountably many such groups to work with. So we have an enormous class of groups to study and to consider as potential counter examples for long stood conjectures.

Groups of type FP had been constructed before (the first in 1990), however the techniques used in each previous construction all relied on the same methods. Namely Morse theory on cubical complexes. My work has developed a new construction, the first to not use Morse theory or cubical complexes. We use Gromov's graphical small cancellation theory and finite polygonal cell complexes.

This has given us a new uncountable family of groups of type FP that were not known before. The properties that this new family posses give hope of being able to better understand 'almost finitely presented groups' and to continue to develop the area.

In the process of delivering my results I have developed the literature in the areas of small cancellation theory (a powerful combinatorial tool in group theory) and the area of non-trivial acyclic CW-complexes (interesting spaces first considered by Poincare). These are extremely useful in my work and they will continue to be useful throughout other areas of mathematics.
Exploitation Route Many researchers across the globe study the properties of groups of type FP. This new family gives them a brand new input with which to develop their own work.
Sectors Other

 
Description Attending conferences and meetings. 
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 Throughout my PhD I have attended workshops and conferences regularly. These include YGGT 7 (Switzerland), YGGT 8 (Bilbao), YGGT 9 (Rennes), multiple occurrences of BiSh (Bielefeld and Southampton), GGSE (UCL, Cambridge, Oxford, Southampton) as well as various stand alone events across the UK and the rest of Europe.

I feel these meetings are vital in order to discuss you work with other leading experts across the world.
Year(s) Of Engagement Activity 2018,2019,2020
 
Description Discussion session 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Postgraduate students
Results and Impact Presented my work at a conference in Bilbao - YGGT IIX. I gave an insight into the tools used in my work, this involved a talk followed by a discussion session which increased the output of my work.
Year(s) Of Engagement Activity 2019
 
Description Invited speaker 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach National
Primary Audience Postgraduate students
Results and Impact Presented my work to the mathematics department in Bristol. This resulted in discussion about my work and greatly developed my own progress as well as giving an insight to the attendees.
Year(s) Of Engagement Activity 2019
 
Description PGR Seminar speaker 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Postgraduate students
Results and Impact I presented my work to the rest of the faculty. This resulted in a question session and a deeper understanding of a topic to both myself and my audience.

This occurred five times during my PhD and I am planning to give at least one more before I submit.
Year(s) Of Engagement Activity 2018,2019
 
Description YRM 2018 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach National
Primary Audience Postgraduate students
Results and Impact I was part of a small team of postgraduate students who organised the conference YRM 2018. My main role was financial; I contacted businesses and worked with them in order to secure funding for the conference as well as working out how the conference would best suit their own needs. We secured more funding than ever previously and were, as such, able to cover all attendees costs as well as providing a conference dinner and food across the entire week.

My secondary role involved chairing the talks. This involved keeping the speakers and the attendees happy and involved, as well as maintaining the discussion post talk in order to stimulate the workplace.

We also included a public lecture which had nearly 200 members of public attending. This was a great success.
Year(s) Of Engagement Activity 2018
URL https://yrm2018.wordpress.com