# Research Group — Zero-Dimensional Symmetry

Date Time Room Speaker Title
27.03.2019 10.00 W 243 Yossi Bokor What doughnuts tell us about data
The old joke is that a topologist can't distinguish between a coffee cup and a doughnut. A recent variant of Homology, called Persistent Homology, can be used in data analysis to understand the shape of data. I will give an introduction to persistent Homology and describe two example applications of this tool.

05.02.2019 12.00 MC G29 Alastair Anderberg, Max Carter, Peter Groenhout
William Roland-Batty, Chloe Wilkins
Summer Projects Dress Rehearsal

18.12.2018 14.00 MC LG17 Max Carter, Peter Groenhout Summer Projects

11.12.2018 14.00 MC G29 Davide Spriano Convexity and generalization of hyperbolicity
Almost by definition, the main tool and goal of Geometric Group Theory is to find and exploit correspondences between geometric and algebraic features of groups. Following this philosophy, I will focus on the question: what does it mean for a sub(space/group) to "sit nicely" inside a bigger (space/group)? Focusing on groups, for a subgroup H of a group G, possible answers for the above question are when the subgroup H is: quasi-isometrically embedded, undistorted, normal/malnormal, finitely generated, geometrically separated...
Many of the above are equivalent when H is a quasiconvex subgroup of a hyperbolic group G, providing very successful correspondences between geometric and algebraic properties of subgroups.
The goal of this talk is to review quasiconvexity in hyperbolic spaces and try to generalize several of those features in a broader setting, namely the class of hierarchically hyperbolic groups (HHG). This is a joint work with Hung C. Tran and Jacob Russell.

04.12.2018 all week Adelaide AustMS Meeting

27.11.2018 14.00 MC LG17 Alejandra Garrido Hausdorff dimension and normal subgroups of free-like pro-$p$ groups
Hausdorff dimension has become a standard tool to measure the "size" of fractals in real space. However, it can be defined on any metric space and therefore can be used to measure the "size" of subgroups of, say, pro-$p$ groups (with respect to a chosen metric). This line of investigation was started 20 years ago by Barnea and Shalev, who showed that $p$-adic analytic groups do not have any "fractal" subgroups, and asked whether this characterises them among finitely generated pro-$p$ groups.
I will explain what all of this means and report on joint work with Oihana Garaialde and Benjamin Klopsch in which, while trying to solve this problem, we ended up showing an analogue of a theorem of Schreier in the context of pro-$p$ groups of positive rank gradient: any finitely generated infinite normal subgroup of a pro-$p$ group of positive rank gradient is of finite index. I will also explain what "positive rank gradient" means, and why pro-$p$ groups with such a property are "free-like".

20.11.2018 14.00 MC 110 Thibaut Dumont Cocycles on trees and piecewise translation action on locally compact groups
In the first part of this seminar, I will present some geometric cocycles associated to trees and ways to compute their norms. Similar construction exists for Euclidean buildings but no satisfactory estimates of the norm is currently known. In the second part, I will discuss some ongoing research with Thibaut Pillon on actions the infinite cyclic group by piecewise translations on locally compact group. Piecewise translation actions have been well studied for finitely generated groups, e.g. by Whyte, and provide positive answers to the von-Neumann-Day problem or the Burnside problem. The generalization to LC-groups was introduced by Schneider. The topic seems to have interesting implications for tdlc-groups.

13.11.2018 all day X 602 EViMS Workshop

12.11.2018 14.00 MC G29 Anne Thomas Divergence in right-angled Coxeter groups
The divergence of a pair of geodesics in a metric space measures how fast they spread apart. For example, in Euclidean space all pairs of geodesics diverge linearly, while in hyperbolic space all pairs of geodesics diverge exponentially. In the 1980s Gromov proved that in symmetric spaces of non-compact type, the only possible divergence rates are linear or exponential, and he asked whether the same dichotomy holds in CAT(0) spaces. Soon afterwards, Gersten used these ideas to define a quasi-isometry invariant, also called divergence, which measures the "worst" rate of divergence. Gersten and others have since found many examples of finitely generated groups with quadratic divergence. We study divergence in right-angled Coxeter groups with triangle-free defining graphs. Using the structure of certain flats in the associated Davis complex, which is a CAT(0) square complex, we characterise such groups with linear and quadratic divergence, and construct examples of right-angled Coxeter groups with divergence polynomial of arbitrary degree. This is joint work with Pallavi Dani (Louisiana State University).

06.11.2018 all day U Sydney Group Actions Seminar held at the University of Sydney

30.10.2018 14.00 MC LG17 Reading Group

23.10.2018 all day U Sydney Group Actions Seminar held at the University of Sydney

16.10.2018 14.00 MC LG17 Alejandra Garrido Maximal subgroups of some groups of intermediate growth
Given a group one of the most natural things one can study about it is its subgroup lattice, and the maximal subgroups take a prominent role. If the group is infinite, one can ask whether all maximal subgroups have finite index or whether there are some (and how many) of infinite index. After telling some historical developments on this question, I will motivate the study of maximal subgroups of groups of intermediate growth and report on joint work with Dominik Francoeur where we give a complete description of all maximal subgroups of some "siblings" of Grigorchuk's group.

09.10.2018 14.00 MC LG17 Dave Robertson Algebraic theory of self-similar groups
I will describe the relationship between self-similar groups, permutational bimodules and virtual group endomorphisms. Based on chapter 2 of Nekrashevych’s book.

02.10.2018 14.00 MC LG17 Alex Bishop The Group Co-Word Problem
In this talk, we will introduce a class of tree automorphism groups known as bounded automata. From this definition, we will see that many of the interesting examples of self-similar groups in the literature are members of this class.
A problem in group theory is classifying groups based on the difficulty of solving their co-word problems, that is, classifying them by the computational difficulty to decide if a word is not equivalent to the identity. Some well-known results in this study are that a group has a co-word problem given by a regular language if and only if it is finite, a deterministic context-free language if and only if it is virtually free, and a deterministic one-counter machine if and only if it is virtually cyclic. Each of these language classes corresponds to a natural and well-studied model of computation.
We will show that the class of bounded automata groups has a co-word problem given by an ET0L language – a class of formal language which has recently gained popularity in areas of group theory. This strengthens a recent result of Holt and Röver (who showed this result for a less restrictive class of language) and extends a result of Ciobanu-Elder-Ferov (who proved this result for the first Grigorchuk group).

25.09.2018 14.00 MC LG17 Timothy Bywaters Spaces at infinity for hyperbolic totally disconnected locally compact groups
Every compactly generated t.d.l.c. group acts vertex transitively on a locally finite graph with compact open vertex stabilisers. Such a graph is called a rough Cayley graph and, up to quasi-isometry, is an invariant for the group. This allows us to define Gromov hyperbolic t.d.l.c. groups and their Gromov boundary in a way analogous to the finitely generated case. The space of directions of a t.d.l.c. group is a metric space 'at infinity' obtained by analysing the action of the group on the set of compact open subgroups. It is particularly useful for detecting flat subgroups, think subgroups that look like $\mathbb{Z}^n$.
In my talk, I will introduce these two concepts of boundary and give some new results which relate them. Time permitting, I may also give details about the proofs.

10.09.2018 14.00 MC G29 Colin Reid Endomorphisms of profinite groups
Given a profinite group $G$, we can consider the semigroup $\mathrm{End}(G)$ of continuous homomorphisms from $G$ to itself. In general $\lambda \in\mathrm{End}(G)$ can be injective but not surjective, or vice versa: consider for instance the case when $G$ is the group $F_p[[t]$ of formal power series over a finite field, $n$ is an integer, and $\lambda_n$ is the continuous endomorphism that sends $t^k$ to $t^{k+n}$ if $k+n \ge 0$ and $0$ otherwise. However, when $G$ has only finitely many open subgroups of each index (for instance, if $G$ is finitely generated), the structure of endomorphisms is much more restricted: given $\lambda \in\mathrm{End}(G)$, then $G$ can be written as a semidirect product $N \rtimes H$ of closed subgroups, where $\lambda$ acts as an automorphism on $H$ and a contracting endomorphism on $N$. When $\lambda$ is open and injective, the structure of $N$ can be restricted further using results of Glöckner and Willis (including the very recent progress that George told us about a few weeks ago). This puts some restrictions on the profinite groups that can appear as a '$V_+$' group for an automorphism of a t.d.l.c. group.

03.09.2018 14.00 MC G29 Stephan Tornier An introduction to self-similar groups
We introduce the notion of self-similarity for groups acting on regular rooted trees as well as their description using automata and wreath iteration. Following the definition of Grigorchuk's group we show that it is an infinite, finitely generated $2$-group. The proof illustrates the use of self-similarity.

27.08.2018 14.00 MC G29 George Willis The tree representation theorem and automorphism groups of rooted trees
(joint work with R. Grigorchuk ad D. Horadam) The tree representation theorem represents a certain group associated with the scale of an automorphism of a t.d.l.c. group as acting by symmetries of a regular (unrooted) tree. It shows that groups acting on regular trees are a fundamental part of the theory of t.d.l.c. groups.
There is also an extensive theory of self-similar and self-replicating groups of symmetries of rooted trees which has developed from the discovery (or creation) of examples such as the Grigorchuk groups.
It will be seen in this talk that these two branches of research are studying essentially the same groups.

20.08.2018 14.00 MC G29 George Willis Locally pro-p contraction groups are nilpotent
A contraction group is a pair $(G,\alpha)$ in which $G$ is a locally compact group and $\alpha$ is an automorphism of $G$ such that $\alpha^n(x)\to 1$ as $n\to\infty$. In joint work with H. Glöckner, it is shown that every contraction group is the direct sum of closed subgroups $$G = D\oplus T$$ with $D$ divisible, i.e. for every $x\in D$ and $n>0$ there is $y\in D$ with $y^n =x$ and $T$ torsion, i.e. there is $n>0$ such that $x^n = 1$ for every $x\in T$. Furthermore, $D$ is the direct sum $$D = \bigoplus_{i=1}^k D_{p_i}$$ of $p_i$-adic analytic nilpotent contraction groups for some prime numbers $p_1,\ldots, p_k$. The torsion subgrou $T$ may also be written as a composition series of simple contraction groups. In the case when all the composition factors are of the form $\mathbb{F}_p(\!(t)\!), \alpha$ with $\alpha$ being the automorphism of multiplication by $p$, it follows easily that $G$ is a solvable group. These ideas will be explained in the talk and a sketch will be presented of a proof that $G$ is in fact nilpotent in this case.

13.08.2018 14.00 MC G29 Michal Ferov Separating cyclic subgroups in graph products of groups
(joint work with Federico Berlai) A natural way to study infinite groups is via looking at their finite quotients. A subset S of a group G is then said to be (finitely) separable in G if we can recognise it in some finite quotient of G, meaning that for every g outside of S there is a finite quotient of G such that the image of g under the canonical projection does not belong to the image of S. We can then describe classes of groups by specifying which types of subsets do we require to be separable: residually finite groups have separable singletons, conjugacy separable groups have separable conjugacy classes of elements, cyclic subgroup separable groups have separable cyclic subgroups and so on... We could also restrict our attention only to some class of quotients, such as finite p-groups, solvable, alternating... Properties of this type are called separability properties. In case when the class of admissible quotients has reasonable closure properties we can use topological methods.
We prove that the property of being cyclic subgroup separable, that is having all cyclic subgroups closed in the profinite topology, is preserved under forming graph products.
Furthermore, we develop the tools to study the analogous question in the pro-p case. For a wide class of groups we show that the relevant cyclic subgroups - which are called p-isolated - are closed in the pro-p topology of the graph product. In particular, we show that every p-isolated cyclic subgroup of a right-angled Artin group is closed in the pro-p topology and, consequently, we show that maximal cyclic subgroups of a right-angled Artin group are p-separable for every p.

06.08.2018 14.00 MC G29 Stephan Tornier Totally disconnected, locally compact groups from transcendental field extensions
(joint work with Timothy Bywaters) Let E over K be field extension. Then the group of automorphisms of E which pointwise fix K is totally disconnected Hausdorff when equipped with the permutation topology. We study examples, aiming to establish criteria for this group to be locally compact, non-discrete and compactly generated.

30.07.2018 14.00 MC G29 Ben Brawn Voltage and derived graphs and their relation to the free product of graphs
We look at a classical construction known as ordinary voltage graphs and their derived graphs. We show how to construct the free product of graphs as the derived graph of a voltage graph whose base graph is the Cartesian product of the given graphs with a specific voltage assignment. We find that the voltage group is always a free group and give the number of generators needed.

23.07.2018 14.00 MC G29 Colin Reid A lemma for group actions on zero-dimensional spaces
I present a lemma concerning a group action on a locally compact zero-dimensional spaces, where the group has a 'small' (compact, say) generating set, relating invariant compact sets with orbit closures. A typical example to have in mind is a compactly generated tdlc group acting on itself by conjugation, where we use conditions on closures of conjugacy classes to deduce the existence of compact normal subgroups. The idea of the lemma has appeared several times in the literature but does not appear to have been given explicitly in this form. I will discuss various applications depending on time.

12.06.2018 14.00 V 206 Dave Robertson Topological full groups - Part II
For an action of a group G on the Cantor set, we can construct a group of transformations of the Cantor set that are constructed 'piecewise' from elements of G. This is called the topological full group of G. Examples include the topological full groups associated to a minimal homeomorphism of the Cantor set considered by Giordano, Putnam and Skau, and Neretin's group of spheromorphisms. I will describe the construction using groupoids, and show how certain examples admit a totally disconnected locally compact topology. This is based on work in progress with Alejandra Garrido and Colin Reid.

05.06.2018 14.00 V 206 Dave Robertson Topological full groups - Part I
For an action of a group G on the Cantor set, we can construct a group of transformations of the Cantor set that are constructed ' piecewise' from elements of G. This is called the topological full group of G. Examples include the topological full groups associated to a minimal homeomorphism of the Cantor set considered by Giordano, Putnam and Skau, and Neretin's group of spheromorphisms. I will describe the construction using groupoids, and show how certain examples admit a totally disconnected locally compact topology. This is based on work in progress with Alejandra Garrido and Colin Reid.

29.05.2018 14.00 V 206 Michal Ferov Profinite words and inverse limits of finite state automata
In the case of finitely generated discrete groups, the problem of deciding whether a product of a sequence of generators and their inverses represents the trivial element is known as the word problem. Somewhat surprisingly, the complexity of word problem is tightly connected to the structure and geometry of the group: a classical result of Anisimov states that a group has word problem decidable by finite-state automaton if and only if the group is finite; similarly, result of Muller and Shupp states that a group has word problem is decidable by push-down automaton if and only if the group is virtually-free. In my talk, I will define inverse limits of finite-state automata and discuss how it might be useful for studying totally-disconnected locally-compact groups.

22.05.2018 14.00 V 206 Thomas Murray On automorphism groups of regular rooted groups
Starting with the automorphism group of a regular, locally finite tree the tree representation theorem leads us to groups acting on a regular, rooted tree. Furthermore these groups satisfy a property called R and are profinite. As a result, the study of these groups may be reduced to those that act on a finite depth regular rooted tree with corresponding finite version of property R. We introduce the idea of studying such groups with geometric objects in order to study trees of higher valency and investigate conjectures made for the binary rooted tree.

15.05.2018 14.00 LSTH 100 George Willis School Seminar — Zero-Dimensional Symmetry
The pleasure and utility of observing symmetry in nature may be found in the mathematics of symmetry, which is known as group theory. Zero-dimensional symmetry is the symmetry of networks and relationships, such as a family tree. In contrast, physical objects, such as a sphere, have positive-dimensional symmetry. While positive-dimensional symmetry has been well understood for more than a century (and is applied in physics) it is only in the last 25 years that our understanding of zero-dimensional symmetry has begun to catch up. Even though great progress is being made, we still aren’t sure how close we are to having the full picture.

08.05.2018 14.00 V 206 Thomas Taylor Automorphisms of Cayley graphs for right-angled Artin groups

01.05.2018 14.00 V 126 George Willis Project — Zero-Dimesional Symmetry
The project on 0-dimensional symmetry, that is, totally disconnected locally compact groups, is organised around four themes, namely, ‘Structure theory’, ‘Geometries’, ‘Local structure and commensurators’ and ‘Representations and computation’. These themes relate to the scale function on a t.d.l.c. group as follows. The scale itself is defined directly in terms of commensuration and the tidying procedure enables computation of the scale. Tidy subgroups can also be characterised geometrically, and the scale behaves naturally under structural decompositions of groups.

24.04.2018 14.00 V 205 Ben Brawn On quasi-label-regular trees and their classification
We introduce almost isomorphisms of locally-finite infinite graphs and in particular trees. We introduce a type of infinite tree, dubbed label-regular, and consider trees that are label-regular except at a finite number of vertices, which we call quasi-label-regular trees. We show how to determine if two quasi-label-regular trees are almost isomorphic or not. We count the number of equivalence classes of quasi-label-regular trees under almost isomorphisms and find this number ranges from finite to infinite.

17.04.2018 14.00 V 205 Stephan Tornier Groups acting on trees with non-trivial quasi-center
We highlight the role of the quasi-center of a t.d.l.c. group in Burger-Mozes theory and present new results concerning the types of automorphisms that the quasi-center of a non-discrete subgroup of the automorphism group of a regular tree may contain in terms of its local action. A theorem which shows that said result is sharp is also presented. We include a proof of the fact that a non-discrete, locally transitive subgroup of the automorphism group of a regular tree does not contain a quasi-central involution.

10.04.2018 14.00 V 205 Stephan Tornier An introduction to Burger-Mozes theory
We recall the types of automorphisms of trees and introduce the notion of local action. After an excursion into permutation group theory, specifically the notions of transitivity, semiprimitivity, quasiprimitivity, primitivity and 2-transitivity, we give an introduction to Burger-Mozes theory of closed, non-discrete subgroups of the automorphism group of a (regular) tree which are locally quasiprimitive.

05.04.2018 14.00 V 205 Colin Reid Totally disconnected, locally compact groups
I give an overview of totally disconnected, locally compact (t.d.l.c.) groups: what they are and in what contexts they arise. In particular, t.d.l.c. groups encompass many classes of automorphism groups of structures, and also occur as completions of groups that have commensurated subgroups. I then discuss some techniques and approaches for studying them, particularly with an eye to general structural questions, and the recent progress that has been made.
We investigate the structure of topological groups and, in particular, symmetry groups of discrete structures such as networks, combinatorial graphs and complexes. The topology on these symmetry groups is locally compact and totally disconnected; such spaces are also called 0-dimensional. The overall goal of our research is to achieve a complete description of 0-dimensional groups that says how a given group may be broken down into simpler ones; classifies the simple ones; relates the global structure of the groups to their local structure; and says how they may be represented geometrically, algebraically and computationally.

Our research may be contrasted with theory of connected locally compact groups, which was largely completed in the 1950s with the solution of Hilbert's Fifth Problem. Building on the earlier foundational work of Sophus Lie, Elie Cartan, Emmy Noether and Hermann Weyl, this theory was vital in twentieth century physics because 4-dimensional space-time, the equations of the quantum atom, and the strong nuclear force all have connected symmetry groups. Whereas they are symmetry groups of physical space, the totally disconnected groups we seek to understand are the symmetry groups of networks, or cyberspace.

Experimental calculations with finite groups guide many aspects of this research. Our longer term aim is to convert theorems into algorithms and to develop computational and visualisation tools for infinite groups which will be made available online. Since the groups are topological, using approximation to reduce the calculations to finite groups will be an important aspect of achieving that aim (p-adic analysis is a particular but restricted case where this approximation is understood).

0-dimensional groups have links with combinatorics, number theory, finite and profinite group theory, geometric group theory, Lie groups, harmonic analysis, descriptive set theory and logic. International research on 0-dimensional groups has grown rapidly in the past decade or so and much of this new activity has been stimulated by breakthroughs made at the University of Newcastle. Our researchers continue to make ground-breaking advances in collaboration with mathematicians from America, Europe and Asia. This research is being supported by Australian Research Council funds of $2.8 million in the period 2018-22.  Ben Brawn Michal Ferov Alejandra Garrido Thomas Murray Colin Reid Dave Robertson Thomas Taylor Stephan Tornier George Willis ### April 2019 ## Special Semester at the Bernoulli Center ### Our research group was recently awarded a special semester at the Bernoulli Center in Lausanne, Switzerland. The semester entitled "Locally compact groups acting on discrete structures" is to take place in the second half of 2020 and will comprise a summer school, several workshops as well as special lectures, bringing together researchers in the field across all academic levels. ### November 2018 ## PhD Scholarship ### A PhD scholarship opportunity is available for students to investigate totally disconnected, locally compact groups under the supervision of ARC Laureate Professor George Willis. The students will join a team seeking to bring our understanding of these groups to a level comparable to that of finite. ### October 2018 ## 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.  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. # Symmetry in Newcastle (SIN) This is a series of meetings with the aim of bringing together mathematicians working on Symmetry -- broadly understood -- that are based around Newcastle -- also broadly understood. Topics of interest include all aspects of group theory and connections to computer science, dynamics, graph theory, logic, number theory, operator algebras, topology. All interested are warmly invited to attend. It's also a great opportunity to discover why Newcastle is Australia's coolest coastal town! The mailing list carries updates and reminders as new seminars are announced. To be added to or removed from the mailing list, or for any other information, please contact Alejandra Garrido. Getting there: The venue (search for room) on the Callaghan campus of The University of Newcastle can be reached in at least three ways: By bus, going to "Mathematics Building, Ring Rd"; by train, going to "Warabrook Station" and walking about 15-20 minutes across the campus; or by car, parking e.g. in carpark "P2". ## Upcoming Events Date Time Room Speaker Title 03.05.2019 12.00-13.00 TBA Heiko Dietrich Quotient algorithms (a.k.a. how to compute with finitely presented groups) In this talk, I will survey some of the famous quotient algorithms that can be used to compute efficiently with finitely presented groups. The last part of the talk will be about joint work with Alexander Hulpke (Colorado State University): we have looked at quotient algorithms for non-solvable groups, and I will report on the findings so far. 14.00-15.00 TBA Youming Qiao TBA 15.30-16.30 TBA Nicole Sutherland Computations of Galois groups and splitting fields The Galois group of a polynomial is the automorphism group of its splitting field. These automorphisms act by permuting the roots of the polynomial so that a Galois group will be a subgroup of a symmetric group. Using the Galois group the splitting field of a polynomial can be computed more efficiently than otherwise, using the knowledge of the symmetries of the roots. I will present an algorithm developed by Fieker and Klueners, which I have extended, for computing Galois groups of polynomials over arithmetic fields as well as approaches to computing splitting fields using the symmetries of the roots. ## Past Events  05.04.2019 12.00-13.00 W 238 Arnaud Brothier Jones' actions of the Thompson's groups: applications to group theory and mathematical physics Motivating in constructing conformal field theories Jones recently discovered a very general process that produces actions of the Thompson groups$F$,$T$and$V$such as unitary representations or actions on$C^{\ast}$-algebras. I will give a general panorama of this construction along with many examples and present various applications regarding analytical properties of groups and, if time permits, in lattice theory (e.g. quantum field theory). 14.00-15.00 W 238 Lawrence Reeves An irrational-slope Thompson's group Let$t$be the the multiplicative inverse of the golden mean. In 1995 Sean Cleary introduced the irrational-slope Thompson's group$F_t$, which is the group of piecewise-linear maps of the interval$[0,1]$with breaks in$Z[t]$and slopes powers of$t$. In this talk we describe this group using tree-pair diagrams, and then demonstrate a finite presentation, a normal form, and prove that its commutator subgroup is simple. This group is the first example of a group of piecewise-linear maps of the interval whose abelianisation has torsion, and it is an open problem whether this group is a subgroup of Thompson's group$F$. 15.30-16.30 W 238 Richard Garner Topos-theoretic aspects of self-similarity A Jonsson-Tarski algebra is a set$X$endowed with an isomorphism$X\to XxX$. As observed by Freyd, the category of Jonsson-Tarski algebras is a Grothendieck topos - a highly structured mathematical object which is at once a generalised topological space, and a generalised universe of sets. In particular, one can do algebra, topology and functional analysis inside the Jonsson-Tarski topos, and on doing so, the following objects simply pop out: Cantor space; Thompson's group V; the Leavitt algebra L2; the Cuntz semigroup S2; and the reduced$C^{\ast}-algebra of S2. The first objective of this talk is to explain how this happens. The second objective is to describe other "self-similar toposes" associated to, for example, self-similar group actions, directed graphs and higher-rank graphs; and again, each such topos contains within it a familiar menagerie of algebraic-analytic objects. If time permits, I will also explain a further intriguing example which gives rise to Thompson's group F and, I suspect, the Farey AF algebra. No expertise in topos theory is required; such background as is necessary will be developed in the talk.

In this edition, three 2017 ARC Laureate Fellows in mathematics outline their projects. Descriptions of these projects are given as abstracts below.

 15.03.2019 12.00-13.00 Purdue Room Mathai Varghese Advances in Index Theory The project aims to develop novel techniques to investigate Geometric analysis on infinite dimensional bundles, as well as Geometric analysis of pathological spaces with Cantor set as fibre, that arise in models for the fractional quantum Hall effect and topological matter, areas recognised with the 1998 and 2016 Nobel Prizes. Building on the applicant's expertise in the area, the project will involve postgraduate and postdoctoral training in order to enhance Australia's position at the forefront of international research in Geometric Analysis. Ultimately, the project will enhance Australia's leading position in the area of Index Theory by developing novel techniques to solve challenging conjectures, and mentoring HDR students and ECRs. 14.00-15.00 Purdue Room Fedor Sukochev Breakthrough methods for noncommutative calculus This project aims to solve hard, outstanding problems which have impeded our ability to progress in the area of quantum or noncommutative calculus. Calculus has provided an invaluable tool to science, enabling scientific and technological revolutions throughout the past two centuries. The project will initiate a program of collaboration among top mathematical researchers from around the world and bring together two separate mathematical areas into a powerful new set of tools. The outcomes from the project will impact research at the forefront of mathematical physics and other sciences and enhance Australia's reputation and standing. 15.30-16.30 Purdue Room George Willis Zero-Dimensional Symmetry and its Ramifications This project aims to investigate algebraic objects known as 0-dimensional groups, which are a mathematical tool for analysing the symmetry of infinite networks. Group theory has been used to classify possible types of symmetry in various contexts for nearly two centuries now, and 0-dimensional groups are the current frontier of knowledge. The expected outcome of the project is that the understanding of the abstract groups will be substantially advanced, and that this understanding will shed light on structures possessing 0-dimensional symmetry. In addition to being cultural achievements in their own right, advances in group theory such as this also often have significant translational benefits. This will provide benefits such as the creation of tools relevant to information science and researchers trained in the use of these tools. 01.03.2019 12.00-13.00 W 104 Marcelo Laca An introduction to KMS states and two suprising examples The KMS condition for equilibrium states of C*-dynamical systems has been around since the 1960's. With the introduction of systems arising from number theory and from semigroup dynamics following pioneering work of Bost and Connes, their study has accelerated significantly in the last 25 years. I will give a brief introduction to C*-dynamical systems and their KMS states and discuss two constructions that exhibit fascinating connections with key open questions in mathematics such as Hilbert's 12th problem on explicit class field theory and Furstenberg's x2 x3 conjecture. 14.00-15.00 W 104 Zahra Afsar KMS states of $C^*$-algebras of $*$-commuting local homeomorphisms and applications in $k$-graph algebras In this talk, I will show how to build $C^*$-algebras using a family of local homeomorphisms. Then we will compute the KMS states of the resulted algebras using Laca-Neshveyev machinery. Then I will apply this result to $C^*$-algebras of $K$-graphs and obtain interesting $C^*$-algebraic information about $k$-graph algebras. This talk is based on a joint work with Astrid an Huef and Iain Raeburn. 15.30-16.30 W 104 Aidan Sims What equilibrium states KMS states for self-similar actions have to do with fixed-point theory Using a variant of the Laca-Raeburn program for calculating KMS states, Laca, Raeburn, Ramagge and Whittaker showed that, at any inverse temperature above a critical value, the KMS states arising from self-similar actions of groups (or groupoids) $G$ are parameterised by traces on C*(G). The parameterisation takes the form of a self-mapping $\chi$ of the trace space of C*(G) that is built from the structure of the stabilisers of the self-similar action. I will outline how this works, and then sketch how to see that $\chi$ has a unique fixed-point, which picks out the preferred" trace of C*(G) corresponding to the only KMS state that persists at the critical inverse temperature. The first part of this will be an exposition of results of Laca-Raeburn-Ramagge-Whittaker. The second part is joint work with Joan Claramunt.
Date Time Room Speaker Title
06.06.2018 14.00 V 206 George Willis Free products of graphs - Part II

30.05.2018 14.00 V 206 George Willis Free products of graphs - Part I
Date Time Room Speaker Title
TBA 10.00 W 243 Dave Robertson Essentially Chief Series

10.04.2019 10.00 W 243 Dave Robertson Essentially Chief Series

08.04.2019 13.00 W 243 Dave Robertson Essentially Chief Series

01.04.2019 13.00 W 243 Stephan Tornier Cayley-Abels graphs

20.03.2019 10.00 W 243 Stephan Tornier Cayley-Abels graphs

18.03.2019 13.00 W 243 Alejandra Garrido Cayley-Abels graphs

13.03.2019 10.00 W 243 Alejandra Garrido Cayley-Abels graphs

11.03.2019 13.00 W 243 Alejandra Garrido Cayley-Abels graphs

06.03.2019 10.00 W 243 Alejandra Garrido Cayley-Abels graphs

04.03.2019 13.00 W 243 Michal Ferov Cayley-Abels graphs

27.02.2019 10.00 W 243 Michal Ferov Cayley-Abels graphs

25.02.2019 13.00 W 243 Michal Ferov Cayley-Abels graphs

20.02.2019 13.00 SR 118 Michal Ferov Cayley-Abels graphs

17.12.2018 14.00 MC G29 Stephan Tornier Haar measures

10.12.2018 14.00 MC G29 Dave Robertson Haar measures

04.12.2018 all week Adelaide AustMS Meeting

03.12.2018 all week Adelaide AustMS Meeting

27.11.2018 9.00 MC G29 Dave Robertson Haar measures

26.11.2018 14.00 MC G29 CARMA Retreat

20.11.2018 9.00 MC G29 Dave Robertson Haar measures

19.11.2018 14.00 MC G29 Dave Robertson Haar measures

13.11.2018 9.00 MC G29 EViMS Workshop

12.11.2018 14.00 MC G29 Weekly Seminar

06.11.2018 9.00 MC G29 Group Actions Seminar held at the University of Sydney

05.11.2018 14.00 MC G29 Exercise Session

30.10.2018 14.00 MC G29 Stephan Tornier Semidirect products and restricted direct products

30.10.2018 9.00 MC G29 Michal Ferov Locally finite graphs

29.10.2018 14.00 MC G29 Michal Ferov Locally finite graphs

22.10.2018 14.00 MC G29 Michal Ferov Topological Isomorphism Theorems

16.10.2018 9.00 MC G29 Michal Ferov Topological structure of t.d.l.c. Polish groups

15.10.2018 14.00 MC G29 Michal Ferov Van Dantzig's theorem
Dates Place Event Participants
December 3-6, 2019 Melbourne,
Australia
Meeting of the Australian Mathematical Society

September 22-27, 2019 Heidelberg,
Germany
Heidelberg Laureate Forum

August 5-9, 2019 Sydney,
Australia
Flags, Galleries and Reflection Groups

May 26-30, 2019 Tel Aviv,
Israel
Geometric and Asymptotic Group Theory
with Applications

April-July, 2019 Będlewo,
Poland
Geometric and Analytic Group Theory Alejandra Garrido

March 24-29, 2019 Dagstuhl,
Germany
Algorithmic Problems in Group Theory Michal Ferov

January 22-25, 2019 Zurich,
Switzerland
Groups, spaces, and geometries
on the occasion of Alessandra Iozzi's 60th birthday
Colin Reid, Stephan Tornier

January 21-25, 2019 Sydney,
Australia
The Asia-Australia Algebra Conference 2019 Michal Ferov, Alejandra Garrido, George Willis

January 16-18, 2019 Auckland,
New Zealand
Groups and Geometries

January 11, 2019 London,
England
Geometric Group Theory meeting at Royal Holloway Colin Reid

Australia
Meeting of the Australian Mathematical Society Alejandra Garrido, Colin Reid, Dave Robertson, Stephan Tornier, George Willis

November 14-16, 2018 Newcastle,
Australia
AMSI-CARMA workshop on Mathematical Thinking George Willis

November 13, 2018 Newcastle,
Australia
Effective Visualisation in the Mathematical Sciences Alejandra Garrido, Colin Reid, Dave Robertson, Stephan Tornier, George Willis

November 9-11, 2018 Newcastle,
Australia
Diagrammatic Reasoning in Higher Education Dave Robertson, Stephan Tornier

September-December, 2018 Bonn,
Germany
Logic and Algorithms in Group Theory Michal Ferov, George Willis

September 21, 2018 London,
England
Hausdorff Dimension Alejandra Garrido

September 11-13, 2018 Geneva,
Switzerland
Spectra and L2 - invariants Alejandra Garrido

September 3-7, 2018 Oxford,
England
Groups, Geometry and Representations Alejandra Garrido

June 25-29, 2018 Düsseldorf,
Germany
Trees, dynamics and locally compact groups Michal Ferov, Alejandra Garrido, Colin Reid, Stephan Tornier, George Willis

June 11-14, 2018 St. Andrews,
England
British Mathematical Colloquium Colin Reid

January 23-26, 2018 Auckland,
New Zealand
Groups and Geometry Michal Ferov, Colin Reid, George Willis

The Group Actions Seminar held regularly at The University of Sydney

The Geometry and Topology Seminar held regularly at The University of Sydney