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

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.