# Research Group — Zero-Dimensional Symmetry

## Summer Projects 2021/22

### The following student research projects are related to the Australian Research Council project on 0-Dimensional Symmetry. While they are independent, each one gives a different view on the overall project. Another, more technical, description of this research is that it concerns totally disconnected, locally compact groups. An overview which attempts to explain the broader research project to non-experts may be seen here. (MFO, CC BY-NC-SA 3.0 )

 Creating and analysing a library of universal groups (Stephan Tornier) Mathematicicans study the intuitive notion of 'symmetry' through the algebraic notion of a 'group'. The symmetry 'groups' of infinite networks, or graphs, constitute a current research frontier and the 'universal groups' appearing in the title form an important class of examples. An existing computational package written in GAP provides methods to find and store universal groups. The goal of this project is to exhaust the capabilities of this package by running its methods on the university's high performance computing grid, organise the results into a library and, time permitting, analyse and visualise this library. Programming experience and basic knowledge of graph theory, combinatorics or linear algebra are essential. Knowledge of algebra (group theory) is desirable. Joining dots to see groups from trees (George Willis) The project involves finding alternative ways of visualising self-similar permutation groups that improve our ability to see patterns among them. Several thousand of these groups, which act on binary trees, have been generated by computer searches. However, the action on the tree is often not illuminating and we seek geometric actions that convey more visual information. For example, given certain permutations of four points, our understanding of the permutations is greatly enhanced if we see that the four points are in fact the vertices of a square and the permutations come from rotations and reflections of the square. Our goal is to `join the dots’ and see geometric patterns in the computer generated cases we have found currently. The approach will again use a computer search. Knowledge of a second-year linear algebra or third-year algebra course is assumed, and knowledge of computing is desirable. Students will learn more algebra and also gain experience working with computer algebra systems. It is possible that work from this project will eventually be included in publicly available computer algebra packages. The project contributes to a much larger one that aims to understand symmetry groups of infinite networks, and students will also learn about this larger project.

## Research Assistant Positions

### 1. Computations with finite graphs

Description: A research assistant is required to work on a project that computes, for small finite graphs, whether the graph can be the 1-sphere of a vertex-transitive graph. The assistant will work under the supervision of Stephan Tornier and George Willis to implement algorithms that check whether a given graph satisfies compatibility conditions that allow it to be extended to a vertex-transitive graph. We aim to build a catalogue of all such graphs up to some size.

Essential: Basic knowledge of graph theory and programming experience.
Desirable: LaTeX

### 2. Computations with self-replicating groups

Description: A research assistant is required to work on a project that searches for automorphism group of finite rooted trees that are quotients of self-replicating groups acting on an infinite tree. The assistant will work under the supervision of Stephan Tornier and George Willis to implement algorithms that find such groups incrementally on trees of increasing depth. At first, this work will be verifying previous calculations but we also aim to extend the range of trees for which these groups are known.

Essential: Group theory (Algebra / MATH3120) and programming experience.
Desirable: LaTeX

## First-Year Summer Projects

### We offer six summer project scholarships around the topic "Puzzles, Codes and Groups" for first year students to be conducted in January and February 2020. Apply to juliane.turner@newcastle.edu.au with your student number. A flyer for the project can be found here.

In this summer project you will explore the mathematical formalisation of the everyday notion of symmetry, which is the algebraic concept of a 'group'. Using groups, we are able to state with certainty whether a given puzzle can be solved (the 15-puzzle can not!) and, if so, compute how many steps are needed. As an example of the far-reaching applicability of this concept, we will look into error-correcting codes, such as the Golay code, which are critical in any digital communication, including NASA's Voyager program. Apart from theoretical studies in the area of groups, possible subprojects include the analysis of a given puzzle, the creation of new ones and the formulation of solution algorithms, as well as the design, analysis and implementation of codes. A team of up to six students will work on this project.

## Summer Projects 2019/20

### The following student research projects are related to the Australian Research Council project on 0-Dimensional Symmetry. While they are independent, each one gives a different view on the overall project. Another, more technical, description of this research is that it concerns totally disconnected, locally compact groups. An overview which attempts to explain the broader research project to non-experts may be seen here. (MFO, CC BY-NC-SA 3.0 )

 Symmetry of Graphs A graph (a network of vertices and edges) is \emph{vertex-transitive} if it 'looks' the same at all vertices. In algebraic terms, this means that, for any pair of vertices, there is an automorphism of the graph which maps the first vertex to the second. A necessary, but not sufficient, condition for a graph to be vertex-transitive is that all vertices should have the same valency. Analysing graph symmetry therefore involves a deeper study of the relationship between graphs and their automorphism groups than vertex valencies. Vertex-transitive graphs need not be edge-transitive, for example, the horizontal edges of a triangular pyramid lie on 3-cycles in the graph but the vertical edges do not. The particular question investigated in this project is how its symmetry group changes as edges are added to, or removed from, a graph. For finite graphs, a classical theorem of Burnside is relevant when the graph has prime order and, for infinite graphs, the goal is to reduce the vertex valency since this number controls important features of the symmetry group. Students taking this project will extend their knowledge of combinatorics and algebra, and how these two topics interact. The project will be jointly supervised by Brian Alspach. Solvable and nilpotent groups The notion of a solvable group originated with the work of É. Galois (1832), who showed that a polynomial equation has a solution by radicals if and only if its group of symmetries is solvable. For example, the formula $x=-b\pm\sqrt{b^{2}-4ac}/2a$ is the solution of a quadratic equation by radicals and symmetries of the equation swap the $+$ and $-$ signs. (A group that is not solvable has some factors which are simple.) Groups of upper triangular $n\times n$ real matrices are solvable and also have the topological property of being connected. It may be shown that these are essentially all the connected solvable groups. This project investigates solvable totally disconnected, locally compact groups. Our starting point is groups of upper triangular matrices having integer entries. These groups have the property of being nilpotent, which is stronger than solvability. Students taking this project will extend their knowledge of algebra, analysis and number theory. 'Totally disconnected' and 'locally compact' are topological notions; 'group', 'solvable and 'nilpotent' are algebraic ones; and the integer matrices embed into matrices over the real numbers as well as over other number fields. Symmetries of Rooted Trees Self-similar symmetry groups of rooted trees contribute to the study of totally disconnected, locally compact (or $0$-dimensional) groups in much the same way as eigenvalues and eigenvectors contribute to linear algebra. Whereas eigenvalues are complex numbers and we have a complete picture of what they all are, we are still at the stage of trying to describe self-similar symmetry groups of rooted trees. It is not necessary to understand the link with the theory of $0$-dimensional symmetry or with eigenvalues in order to describe these groups. This project aims to find an alternative geometric interpretation of some of the symmetry groups of trees. The idea being that describing these geometries might be a more natural approach to describing the groups. Computer algebra software will be used to analyse how the groups act on pairs, triples, etc. of vertices of the trees and then study the polyhedra in which these are edges, faces, etc. Observed patterns in the geometries may be able to be extrapolated to produce new families of groups. Students taking this project will extend their knowledge of algebra, graph theory and mathematical software. 'Group' is an algebraic notion and 'rooted tree' is a combinatorial one which arises in the study of data structures in computer science. 'Totally disconnected' and 'locally compact' are topological notions which, although relevant to the background, are not needed for this project. Visualising $0$-dimensional symmetry The word 'symmetry' brings to mind visual images and geometry. It has a broader meaning in mathematics, where we think of regularly repeating patterns and invariance under transformations as displaying symmetry, and where the language of algebra is used to describe symmetry. Visualising the patterns or the dynamics of the transformations remains an effective tool for understanding the algebra however. This project aims to develop software for visualising various aspects of $0$-dimensional symmetry, which is the symmetry of infinite networks and arises in number theory and other parts of algebra as well. The aim is to produce software which may be used by researchers and which will be made available on web-pages of the $0$-Dimensional Symmetry project. Students taking this project will extend their knowledge of algebra, analysis, mathematical software and coding skills. 'Totally disconnected' and 'locally compact' are topological notions; 'group' is an algebraic one; and other concepts will be met in the course of the project. Groups Acting on Trees: Compatible Local Actions The concept of symmetry is pervasive in mathematics and formalised in the algebraic notion of a 'group'. It is often natural to equip a group with a 'topology' - a generalisation of distance functions - which, in a sense, gives groups a shape. For example, the symmetry groups of infinite networks, or 'graphs', become zero-dimensional. A versatile and accessible class of these groups was defined by Burger-Mozes and refined by this project's supervisor: Picture an infinite graph in which every vertex has the same number of neighbours and consider only those symmetries of the graph which in a neighbourhood of every given vertex act like one of finitely many allowed 'local actions'. In order for the resulting family of symmetries to reflect these restrictions accurately, the local actions need to satisfy certain conditions. The aim of this project is to find more general constructions of such local actions and use them to test the sharpness of an existing rigidity theorem. A student who takes this project will extend his/her knowledge of algebra and learn how to use computer algebra systems designed for computations in group theory, including coding skills. Groups Acting on Trees without Involutive Inversions The concept of symmetry is pervasive in mathematics and formalised in the algebraic notion of a 'group'. It is often natural to equip a group with a 'topology' - a generalisation of distance functions - which, in a sense, gives groups a shape. For example, the symmetry groups of infinite networks, or 'graphs', become zero-dimensional. A versatile and accessible class of these groups was defined by Burger-Mozes and refined by this project's supervisor: Picture an infinite graph in which every vertex has the same number of neighbours and consider only those symmetries of the graph which in a neighbourhood of every given vertex act like one of finitely many allowed 'local actions'. In order for the resulting family of symmetries to reflect these restrictions accurately, the local actions need to satisfy certain conditions. The aim of this project is to define a new class of groups acting on said graph by restricting the local action on edge neighbourhoods rather than vertex neighbourhoods, and thereby gain a new perspective on existing examples of graph symmetry groups relating to the Weiss conjecture. A student who takes this project will extend his/her knowledge of algebra and topology, and develop proofs intertwining both.

## Summer Projects 2018/19

### The following student research projects are related to the Australian Research Council project on 0-Dimensional Symmetry. While they are independent, each one gives a different view on the overall project. Another, more technical, description of this research is that it concerns totally disconnected, locally compact groups. An overview which attempts to explain the broader research project to non-experts may be seen here. (MFO, CC BY-NC-SA 3.0 )

 Simple groups of infinite matrices An essential step towards understanding $0$-dimensional symmetry is to describe the totally disconnected, locally compact (t.d.l.c.) groups which are simple. Simple groups are those which cannot be factored into smaller pieces and they are sometimes called the 'atoms of symmetry', or said to be analogues of the prime numbers in number theory. This project investigates t.d.l.c. groups of infinite matrices. It is suspected that these groups will be found to be simple and we will aim to show that by first studying corresponding groups of $n\times n$ matrices which are known to be simple. Students taking this project will extend their knowledge of algebra, analysis and number theory. 'Totally disconnected', 'locally compact' and '$0$-dimensional' are topological notions; 'group' and 'simple' are algebraic ones; and the matrix entries are numbers modulo a prime number $p$. Solvable and nilpotent groups The notion of a solvable group originated with the work of É. Galois (1832), who showed that a polynomial equation has a solution by radicals if and only if its group of symmetries is solvable. For example, the formula $x=-b\pm\sqrt{b^{2}-4ac}/2a$ is the solution of a quadratic equation by radicals and symmetries of the equation swap the $+$ and $-$ signs. (A group that is not solvable has some factors which are simple.) Groups of upper triangular $n\times n$ real matrices are solvable and also have the topological property of being connected. It may be shown that these are essentially all the connected solvable groups. This project investigates solvable totally disconnected, locally compact groups. Our starting point is groups of upper triangular matrices having integer entries. These groups have the property of being nilpotent, which is stronger than solvability. Students taking this project will extend their knowledge of algebra, analysis and number theory. 'Totally disconnected' and 'locally compact' are topological notions; 'group', 'solvable and 'nilpotent' are algebraic ones; and the integer matrices embed into matrices over the real numbers as well as over other number fields. Free products of graphs Symmetries of networks (or graphs) are '$0$-dimensional', and such symmetries are investigated through the algebraic technique of totally disconnected, locally compact groups. We are interested in highly symmetric, infinite graphs and one way to form such graphs is by gluing together infinitely many copies of finite graphs according to some regular instructions. This project investigates the symmetry groups of examples of graphs formed in this way and compares them with the symmetry groups of infinite regular trees, which are the most basic type of infinite regular graph. The aim is to determine whether the symmetry groups obtained in this way are simple and new. Students taking this project will extend their knowledge of algebra, analysis and combinatorics. 'Totally disconnected' and 'locally compact' are topological notions; 'group' and 'simple' are algebraic ones; and 'graphs' are a combinatorial concept. Visualising $0$-dimensional symmetry The word 'symmetry' brings to mind visual images and geometry. It has a broader meaning in mathematics, where we think of regularly repeating patterns and invariance under transformations as displaying symmetry, and where the language of algebra is used to describe symmetry. Visualising the patterns or the dynamics of the transformations remains an effective tool for understanding the algebra however. This project aims to develop software for visualising various aspects of $0$-dimensional symmetry, which is the symmetry of infinite networks and arises in number theory and other parts of algebra as well. The aim is to produce software which may be used by researchers and which will be made available on web-pages of the $0$-Dimensional Symmetry project. Students taking this project will extend their knowledge of algebra, analysis, mathematical software and coding skills. 'Totally disconnected' and 'locally compact' are topological notions; 'group' is an algebraic one; and other concepts will be met in the course of the project.