Permutation groups, totally disconnected locally compact groups, and the local isomorphism relation.

Lead Research Organisation: University of Lincoln
Department Name: School of Maths and Physics

Abstract

Group theory is the theory of symmetry. It has deep links to all of pure maths, and fundamental applications in physics (e.g. Noether's theorems in general relativity, Wigner's Theorem in quantum mechanics), chemistry (e.g. crystallography) and computer science (e.g. linear algebra in computer graphics and AI, algebraic number theory in cryptography). Developments in group theory precipitate breakthroughs in maths and the sciences. These developments follow from looking at groups from one of a handful of natural perspectives. One such perspective is to view a group as a permutation group - the symmetries of an object. Another is to view a group as a topological object, where the "shape" of the group is studied (the "shape" here is topological and can be stretched and bent, but not cut). An important class of these topological groups are those that are non-discrete and locally compact (these have a nontrivial "shape" that on a local level looks like the space around us) and compactly generated (these groups can be "built" out of local pieces, again like the space around us). Historically, research into these locally compact groups led to important breakthroughs in physics, as well as the development of new areas of mathematics, like abstract harmonic analysis.

The study of locally compact groups breaks into two cases: the connected case and the totally disconnected case. The solution of Hilbert's Fifth problem in the early 1950s led to a broad understanding of the connected case. Understanding the totally disconnected case (henceforth, tdlc) was considered impossible until transformative work by George Willis in the 1990s. Today the study of compactly generated tdlc groups is an important area of research. We now know these groups are strongly related to groups of symmetries (i.e. permutation groups); that they have a geometry, and the interplay between their geometry and topology restricts their structure; and they can be "decomposed" into "simple pieces". A central focus of tdlc theory is to understand these "simple pieces", since they hold the key to understanding the structure of all compactly generated locally compact groups.

In group theory, two groups that are essentially the same are said to be isomorphic. It is known already that we cannot hope to understand these "simple pieces" using the isomorphism relation - the groups are too complicated. However, it is thought that they could be understood using the "local isomorphism" relation, where two groups are locally isomorphic if they have isomorphic "local" (i.e. compact open) subgroups. To understand these "simple pieces" using local isomorphisms, we need as a first step to know how many different (up to local isomorphism) "simple pieces" there are. This is considered to be a very hard and important problem.

At present, no progress can be made - too little is known about local isomorphisms, and there are no general tools available. The proposed research seeks to address this, by exploiting a useful interplay between permutation groups and tdlc theory. The idea is to move the problem into the language of permutation groups and groups acting on graphs, where there are many novel and powerful tools available (some developed recently), solve the problem, and then translate the solution back into the language of compactly generated tdlc groups.

This proposal will lead to a deeper understanding of locally compact groups. This will impact the many areas of maths and physics where locally compact groups are used. The proposal will also increase our understanding of the symmetries and structure of highly-symmetric infinite graphs. These infinite graphs are limiting cases of families of large finite graphs, built from many copies of a smaller graph. These large finite graphs are used extensively in computer science and scientific modelling. This increased understanding could one day lead to more efficient algorithms in computing and scientific modelling.

Publications

10 25 50
 
Title Improvements to research infrastructure - New mathematical technique for finding compactly generated simple tdlc groups using primitive permutation groups with limited scale. 
Description A new mathematical technique has been developed that can detect compactly generated simple tdlc groups inside subdegree-finite closed primitive permutation groups whose scale in some sense grows slowly. 
Type Of Material Improvements to research infrastructure 
Year Produced 2024 
Provided To Others? No  
Impact Too soon to say. At least one research paper will come from this - related to a conjecture of Peter Neumann and Samson Adeleke. The previous best attempt at the conjecture appeared in the highly regarded Journal of the LMS. This new paper will improve considerably on the main result in that paper, so it should appear in a good journal. 
 
Title New mathematical construction technique discovered for creating simple commensurators for profinite groups 
Description In this grant's proposal, "The major challenge" is to answer the following question: Is the number of local isomorphism classes of groups in S uncountable, where S is the class of nondiscrete, compactly generated, topologically simple, tdlc (totally disconnected and locally compact) groups. The PI (Simon Smith) and the PDRA (Ged Corob Cook) both now conjecture that the answer is "yes". To attempt to prove this, our focus has moved on to finding general construction methods for building topologically simple, tdlc commensurators for any permutation representation of any profinite group. We have developed a novel technique for doing this that results in highly unusual infinite simple groups. From this it is possible for us to generate examples with uncountably many local isomorphism classes. Unfortunately, the construction does not guarantee compact generation, so it has not yet supplied the final answer to the major challenge, but a refinement of this technique may do so. 
Type Of Material Improvements to research infrastructure 
Year Produced 2022 
Provided To Others? No  
Impact The impact so far is in using this tool to further the research project funded by this grant. 
 
Description Dr Alejandra Garrido (ICMAT, Spain) visited PI at U Lincoln to discuss topics covered by this grant 8th-10th December 2022 
Organisation Institute of Mathematical Sciences
Country Spain 
Sector Charity/Non Profit 
PI Contribution Dr Alejandra Garrido (ICMAT, Spain) visited PI at U Lincoln to discuss topics covered by this grant 8th-10th December 2022. Alejandra and the PI have discussed research topics prior to this grant, but 2022 was the first time research topics directly related to this grant were discussed.
Collaborator Contribution Dr Alejandra Garrido (ICMAT, Spain) visited PI at U Lincoln to discuss topics covered by this grant 8th-10th December 2022. Alejandra and the PI have discussed research topics prior to this grant, but 2022 was the first time research topics directly related to this grant were discussed.
Impact We were discussing ideas related to constructions of infinite simple groups - it is too soon to say whether or not any of the ideas we generated will be useful to the project.
Start Year 2022
 
Description Dr Colin Reid (U Newcastle, Australia) visited PI to work on topics covered by this grant: 26th-30th September 2022 
Organisation University of Newcastle
Country Australia 
Sector Academic/University 
PI Contribution Dr Colin Reid visited PI to work on topics covered by this grant: 26th-30th September 2022. Several new ideas for tackling problems described in this grant's proposal were generated. We have previously collaborated on other research topics, but 2022 was when we first began discussing ideas related to this grant.
Collaborator Contribution Dr Colin Reid visited PI to work on topics covered by this grant: 26th-30th September 2022. Several new ideas for tackling problems described in this grant's proposal were generated. We have previously collaborated on other research topics, but 2022 was when we first began discussing ideas related to this grant.
Impact Ongoing
Start Year 2022
 
Description Dr Colin Reid visited Lincoln in November 2023 to continue collaboration 
Organisation University of Newcastle
Country Australia 
Sector Academic/University 
PI Contribution Continued to work on several joint projects related to this award proposal.
Collaborator Contribution 50-50 collaboration between the two of us. All ideas are still preliminary.
Impact Still ongoing.
Start Year 2022
 
Description Dr Simon Smith and Dr Ged Corob Cook participated in the multi-centre initiative: "Special semester on totally disconnected locally compact groups" at University of Münster 
Organisation University of Münster
Country Germany 
Sector Academic/University 
PI Contribution Dr Simon Smith and Dr Ged Corob Cook participated in the multi-centre initiative: "Special semester on totally disconnected locally compact groups" at University of Münster. Specifically in the focused event on totally disconnected locally compact groups from a geometric perspective.
Collaborator Contribution - Dr Corob cook gave a talk on his work relating cohomology and tdlc groups - Dr Smith gave a talk on his work relating actions on trees and tdlc groups (an output of this funded project) Both researchers spoke extensively with leading researchers from around the world who also attended the event. Several new connections were made. Our focus was on finding a way through a current bottleneck. Both researchers spoke with PhD students and early career researchers.
Impact Several researchers to visit U Lincoln in 2024 as a result of this: Prof. R. G. Möller (U Iceland) Dr I. Castellano (Bielefeld University, Germany) B. Marchionna (Universities of Milano-Bicocca, Italy)
Start Year 2023
 
Description Ged Corob Cook (PDRA funded by this grant) presented his research (funded by this grant) to politicians at the "STEM for Britain" event at the Houses of Parliament 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach National
Primary Audience Policymakers/politicians
Results and Impact Ged Corob Cook (PDRA funded by this grant) presented his research (funded by this grant) to politicians at the "STEM for Britain" event at the Houses of Parliament on 6th March 2023. Presenting at the event is by application, and Ged's application was accepted. At the event, early career STEM researchers present their research to politicians and civil servants.
Year(s) Of Engagement Activity 2023
 
Description International research conference on "Groups acting on trees" organised and held at U Lincoln. 
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 Conference on "Groups acting on trees" (all talks relevant to this grant) took place at the University of Lincoln on 9th December 2022. Jointly funded by this grant and an award by the London Mathematical Society. Main speakers were Alejandra Garrido ( Universidad Autónoma de Madrid and ICMAT, Spain), Waltraud Lederle (UCLouvain, Belgium) and Naomi Andrew (U Oxford). The conference was hybrid (in-person and online) and was well-attended in-person by postgraduate students from various UK universities. Online attendees included people from Canada, the USA, Belgium, Spain, Sweden and various UK universities.
Year(s) Of Engagement Activity 2022