# Research Group — Zero-Dimensional Symmetry

Date Time Room Speaker Title
10.06.2022 10.30 SR 118 / Zoom Michal Ferov Bounding the complexity of the isomorphism problem for hyperbolic groups; the rigid case

22.04.2022 11.40 SR 118 / Zoom Colin Reid Boundary-2-transitive groups acting on trees that are not locally prosoluble

01.04.2022 12.00 US 321 / Zoom Symmetry in Newcastle

25.03.2022 12.00 SR 118 / Zoom Stephan Tornier Compatibility cocycles of permutation groups

04.03.2022 12.00 US 321 / Zoom Symmetry in Newcastle

25.02.2022 11.30 SR 118 / Zoom João Vitor Pinto e Silva Tits core and elementary rank

18.02.2022 11.30 SR 118 / Zoom George Willis Pairs of roots of flat groups of automorphisms and associated subgroups

11.02.2022 12.00 SR 118 / Zoom Roman Gorazd Higman-Thompson group of directed-label regular trees

06.12.2021 all week Zoom AustMS 2021

29.11.2021 all week Zoom Groups and Geometries

15.11.2021 15.30 Zoom Symmetry in Newcastle

01.11.2021 17.30 Zoom Symmetry in Newcastle

27.09.2021 16.30 Zoom Symmetry in Newcastle

20.09.2021 16.00 Zoom Colin Reid Capillary posets of finite type
In joint work with Alejandra Garrido and Dave Robertson, we have defined a class of posets, called capillary sets, whose almost automorphism groups are a source of examples of topological full groups of homeomorphisms of the Cantor set. Moreover, some homogeneity conditions (recurrent finite type) are enough to ensure that the derived group of $\mathrm{AAut}(\Gamma)$ is a compactly generated simple tdlc group. The problem is to construct new examples, specifically those where the action on the capillary set does not just reduce to looking at almost automorphisms of trees. It turns out that the capillary posets of finite recurrent type are all "finitely presented" in a certain sense (in particular there are only countably many of them), but this still leaves the problem of determining, given a "presentation", whether the poset it produces is actually capillary and of finite recurrent type. I will report on some partial progress on this problem.

13.09.2021 all week Levico Terme (virtual) New Trends around Profinite Groups

06.09.2021 16.00 Zoom Max Carter Representation Theory of Locally Compact Groups: Spherical & Induced Representations
The talk will outline some of the major techniques and results used in the study of representations of general locally compact groups, in particular, techniques and results pertaining to the study of spherical and induced representations will be the focus. Time permitting, I will discuss its relevance to current work on totally disconnected locally compact groups.

31.08.2021 18.00 Zoom Symmetry in Newcastle

24.08.2021 No seminar, World of Group Craft on Thursday and Friday

16.08.2021 all week Banff (virtual) Workshop: Totally Disconnected Locally Compact Groups via Group Actions

09.08.2021 16.30 Zoom Symmetry in Newcastle

26.07.2021 16.00 Zoom João Vitor Pinto e Silva Infinite matrices
To understand a theory, it is always good to work with objects that are easier to understand and work with. With that in mind, me and my supervisors developed a theory for infinite matrices indexed by posets over a ring. On the presentation I will talk about how to define these rings/groups, how to write these groups/rings as inverse limits of finite matrices rings, isomorphism problems and new simple totally disconnected groups that arise from the theory.
This work is a generalization of the work done by Peter Groenhout, Colin D. Reid and George A. Willis, see here.

12.07.2021 16.00 Zoom Sebastian Bischof On the classification of twin buildings
Twin buildings are generalizations of spherical buildings. A natural question is whether it is possible to classify twin buildings. Under mild conditions a local-to-global theorem says that a twin building is determined by its the local structure. Using this theorem one only has to find out which local structures can occur. In this talk I will discuss the current status of the classification and mention some contributions which I made in my PhD project.
Associated arXiv preprint: see here.

05.07.2021 18.00 Zoom Symmetry in Newcastle

28.06.2021 16.00 SR218 / Zoom George Willis Non-locally transitive actions of vertex-transitive groups
The talk will begin with remarks about group actions, generally not locally transitive, on a vertex-transitive graph X. We will then specialise to trees or, from another point of view, lift to the covering tree of X. It will be seen that a general vertex-transitive action may be written as a product of actions that are either locally transitive, as treated by Burger & Mozes, or belong to a complementary class of actions. We will consider whether this complementary class of actions is susceptible to an analysis similar to that of locally transitive actions.

21.06.2021 16.30 Zoom Symmetry in Newcastle

07.06.2021 16.30 Zoom Symmetry in Newcastle

24.05.2021 16.30 Zoom Symmetry in Newcastle

17.05.2021 16.00 Zoom Michal Ferov Automorphisms of Cayley graphs of Coxeter groups I - the case of (non)-discreteness
In an ongoing project with Federico Berlai (University of Vienna) we study automorphism groups of Cayley graphs of Coxeter groups (with respect to the standard generating set). The first step is to identify the ones that are not discrete - we give a characterisation in terms of symmetries of the defining graph. I will sketch out the proof and discuss further directions for research - comments and suggestions will be welcome.

10.05.2021 16.30 Zoom Symmetry in Newcastle

26.04.2021 16.00 Zoom Colin Reid Decomposition of coset spaces in locally compact groups
Phillip Wesolek and I showed a few years ago that every compactly generated locally compact group G admits a closed normal series with finitely many factors, such that each factor is compact, discrete or irreducible as a closed normal factor. It turns out this is a special case of a more general result: given a locally compact second-countable group G and a compactly generated closed subgroup H, there is a finite increasing sequence of closed subgroups going from H to G, such that each group is cocompact, open or maximal closed in the next one (i.e. any subgroup in between is dense in the next group). In the tdlc case a more precise statement can be made in terms of a given compact open subgroup U of G. I will be discussing some work I have done recently around this topic and some thoughts on the problem of understanding maximal closed subgroups of locally compact groups: there are some similarities with the analysis of primitive permutation groups, but also complications due to the possible existence of proper dense subgroups containing the maximal closed subgroup.

19.04.2021 16.30 Zoom Symmetry in Newcastle

12.04.2021 16.00 Zoom George Willis Lattices and contraction groups

29.01.2021 10.00 Zoom Alejandra Garrido Locally compact piecewise full groups - Episode I: General theory
14.00 Zoom Dave Robertson Locally compact piecewise full groups - Episode II: Attack of the capillary sets
15.30 Zoom Colin Reid Locally compact piecewise full groups - Episode III: Revenge of the groups acting on trees

22.01.2021 17.00 Zoom Federico Berlai Cayley graphs of right-angled Coxeter groups
It is known that automorphism groups of locally finite graphs admit a totally disconnected locally compact (tdlc) topology. In this talk I will present some recent results concerning automorphism groups of a particular class of locally finite graphs, that is of Cayley graphs of right-angled Coxeter groups.
Joint work with Michal Ferov.

18.12.2020 all week Zoom Bernoulli Center WinSum School

11.12.2020 all week Zoom AustMS Meeting

23.11.2020 18.30 Zoom Symmetry in Newcastle

09.11.2020 20.00 Zoom Symmetry in Newcastle

05.11.2020 11.00 Zoom George Willis (at Texas A&M) Scale Groups
Scale groups are closed, vertex-transitive groups of automorphisms of a regular tree that fix an end of the tree. These concrete groups emerge from the structure theory of abstract totally disconnected, locally compact groups. There is also a very close correspondence between scale groups and closed self-replicating groups, which are groups of automorphisms of a rooted tree. The role of scale groups in the study of general totally disconnected, locally compact groups and their connection with self-replicating groups will be explained in the talk.

30.10.2020 17.00 Zoom Michal Ferov The conjugacy depth function of wreath products of abelian groups
Informally speaking, the conjugacy depth function of a conjugacy separable group measures how deep in the lattice of normal finite index subgroups one needs to go in order to find a finite quotient witnessing that two elements are not conjugate. The study of conjugacy depth functions is quite new and precise asymptotic bounds are only known for several classes of groups. We study the case when the group in question is a restricted wreath product of finitely generated abelian group. As a consequence we show that the conjugacy depth function for the lamplighter group is exponential. (Ongoing joint work with Mark Pengitore)

23.10.2020 12.15 Zoom Ben Brawn On 1-balls in vertex-transitive graphs

16.10.2020 10.00 Zoom Symmetry in Newcastle

09.10.2020 15.00 Zoom Brian Alspach Recent Progress On Factor-Invariant Graphs
A trivalent vertex-transitive graph X is F(1,2)-invariant if its edge set has a partition into a 2-factor and a 1-factor such that the full automorphism group of X preserves the partition. I shall discuss what we currently know about these graphs. This is joint work with Ted Dobson, Afsaneh Khodadadpour and Don Kreher.

02.10.2020 15.00 Zoom Symmetry in Newcastle

25.09.2020 15.00 Zoom George Willis Flat groups of automorphisms

18.09.2020 15.00 Zoom Symmetry in Newcastle

11.09.2020 15.00 Zoom Stephan Tornier Computational discrete algebra with GAP

04.09.2020 15.00 Zoom Symmetry in Newcastle

28.08.2020 15.00 Zoom João Vitor Pinto e Silva Elementary groups

21.08.2020 15.00 Zoom Symmetry in Newcastle

14.08.2020 15.00 Zoom Colin Reid Pseudo-elementary groups

07.08.2020 15.00 Zoom Symmetry in Newcastle

31.07.2020 15.00 Zoom George Willis Totally disconnected, locally compact groups and the scale
The scale is a positive, integer-valued function defined on any totally disconnected, locally compact (t.d.l.c.) group that reflects the structure of the group. Following a brief overview of the main directions of current research on t.d.l.c. groups, the talk will introduce the scale and describe aspects of group structure that it reveals. In particular, the notions of tidy subgroup, contraction subgroup and flat subgroup of a t.d.l.c. will be explained and illustrated with examples.

24.07.2020 15.00 Zoom Max Carter Cartan decompositions of tdlc groups and two related properties
I will talk about some recent work involving studying Cartan decompositions of tdlc groups. Two closely related properties, the contraction group property and the closed range property will be discussed, along with some applications concerning groups acting on trees.

17.07.2020 15.00 Zoom Symmetry in Newcastle

10.07.2020 15.00 Zoom Michal Ferov Graph automorphisms and colourings
I would like to discuss some ideas I had regarding groups acting on graphs and edge colouring of the said graphs. I will not present any results, but I will point out some questions I have found along the way and I find interesting. Discussion with tips and suggestions will be appreciated.

26.06.2020 15.00 Zoom Symmetry in Newcastle

12.06.2020 15.00 Zoom George Willis Scale groups

05.06.2020 15.00 Zoom Symmetry in Newcastle

29.05.2020 15.00 Zoom Colin Reid Abelian chief factors of locally compact groups

22.05.2020 15.00 Zoom Stephan Tornier On boundary-2-transitive groups acting on trees

16.01.2020 11.00 SR 202 George Willis Scale groups, self-similar groups and self-replicating groups

04.11.2019 10.00 V 108 Max Carter Project Presentation Rehearsal

29.07.2019 11.00 MC LG17 George Willis Computing in automorphism groups of trees

20.05.2019 13.00 W 243 Ben Brawn Automorphisms of forests of quasi-label-regular rooted trees
We investigate when the automorphism group of a quasi-label-regular rooted tree (QLRRT) is trivial or non-trival. We determine when a QLRRT has a finite domain and use this to write its automorphism group as an iterated wreath product. When a QLRRT doesn't have a finite domain, sometimes we can still write its automorphism group as an iterated wreath product and other times we are unlucky and need some number of coupled self-referential equations to describe the group.

08.05.2019 10.00 W 243 Ben Brawn Forests of quasi-label-regular rooted trees and their almost isomorphism classes
We introduce almost isomorphisms of locally-finite 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. When there are a finite number we show how to determine it and suggest a way to choose representatives for the equivalence classes.

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 subgroup $T$ 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.\

06.06.2018 14.00 V 206 George Willis Free products of graphs - Part II

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.

30.05.2018 14.00 V 206 George Willis Free products of graphs - Part I

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.