Research Group — Zero-Dimensional Symmetry
Symmetry in Newcastle 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. 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 and parking, e.g., in carpark "P2". |
Upcoming Events |
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Date | Time | Room | Speaker | Title | |
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20/03/24 | 12.00 - 13.00 | VG 10 / Zoom | Stephen Glasby | Classifying groups with three automorphism orbits | |
We call a group $G$ a $k$-orbit group if its automorphism group $\mathrm{Aut}(G)$ acting naturally on $G$ has precisely $k$ orbits. I will describe the classification of finite $3$-orbit groups after surveying work to classify $k$-orbit groups for small $k$ when $G$ is finite or infinite. The finite $3$-orbit groups that are not $p$-groups are easy to classify. Apart from $Q_8$, the finite non-abelian $3$-orbit $2$-groups are a subset of the Suzuki $2$-groups which Graham Higman [2] classified in 1963. Determining which subset turns out to be far from easy as the automorphism groups of Suzuki $2$-groups are mysterious. Alex Bors and I classified the finite $3$-orbit $2$-groups in [1]. In 2024 Li and Zhu [3], unaware of our work and using different methods, classified the finite groups $G$ where $\mathrm{Aut}(G)$ is transitive on elements of order $p$. Their groups include the $3$-orbit Suzuki $2$-groups, the homocyclic groups $C_{p^n}^m$ of exponent $p^2$ and the generalised quaternion group $Q_{2^{n+1}}$.
I was able to classify all finite 3-orbit groups (including $p>2$) using Hering's Theorem and some representation theory. However, to my surprise Li and Zhu [4] in March 2004 did the same. [1] Alexander Bors and S.P. Glasby, Finite 2-groups with exactly three automorphism orbits, https://arxiv.org/abs/2011.13016v1 (2020). [2] G. Higman, Suzuki 2-groups, Illinois J.~Math. 7 (1963), 79--96. [3] Cai Heng Li and Yan Zhou Zhu, A Proof of Gross' Conjecture on $2$-Automorphic $2$-Groups, https://arxiv.org/abs/2312.16416 (2024). [4] Cai Heng Li and Yan Zhou Zhu, The finite groups with three automorphism orbits, https://arxiv.org/abs/2403.01725 (2024). |
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Past Events |
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25/01/24 | 18.00 - 19.00 | Zoom | Merlin Incerti-Medici | Automorphism groups of cocompact CAT(0) cube complexes | |
Given a cocompact CAT(0) cube complex, we study the group of its cubical isometries, which frequently forms a non-discrete tdlc group. We present a method to study these groups that is focused on our ability to understand the stabilizer subgroups. We demonstrate the potency of this method by introducing a finite, topologically generating set and discuss an important simple subgroup. If there is time, we discuss some open questions regarding the placement of these groups among non-discrete tdlc groups. | |||||
09/08/23 | 14.00 - 15.00 | Zoom | Dilshan Wijesena | Irreducible Pythagorean representations of Thompson’s groups (Slides) | |
Richard Thompson’s groups $F$, $T$ and $V$ are one of the most fascinating discrete infinite groups for their several unusual properties and their analytical properties have been challenging experts for many decades. One reason for this is because very little is known about its representation theory. Luckily, thanks to the novel technology of Jones, a rich family of so-called Pythagorean unitary representation of Thompson’s groups can be constructed by simply specifying a pair of finite-dimensional operators satisfying a certain equality. These representations can even be extended to the celebrated Cuntz algebra and carry a powerful diagrammatic calculus which we use to develop techniques to study their properties. This permits to reduce very difficult questions concerning irreducibility and equivalence of infinite-dimensional representations into problems in finite-dimensional linear algebra. This provides a new rich class of irreducible representations of $F$. Moreover, we introduce the Pythagorean dimension which is a new invariant for all representations of the Cuntz algebra and Pythagorean representations of $F,T,V$. For each dimension $d$, we show the irreducible classes form a moduli space of a real manifold of dimension $2d^2+1$. | |||||
25/05/23 | 16.00 - 17.00 | Zoom | Michael Wibmer | Difference algebraic groups | |
Difference algebraic groups are a generalization of algebraic groups. Instead of just algebraic equations, one allows difference algebraic equations as the defining equations. Here one can think of a difference equation as a discrete version of a differential equation. Besides their intrinsic beauty, one of the main motivations for studying difference algebraic groups is that they occur as Galois groups in certain Galois theories. This talk will be an introduction to difference algebraic groups. | |||||
17.30 - 18.30 | Zoom | Michael Wibmer | Expansive endomorphisms of profinite groups | ||
Étale algebraic groups over a field $k$ are equivalent to finite groups with a continuous action of the absolute Galois group of $k$. The difference version of this well-know result asserts that étale difference algebraic groups over a difference field $k$ (i.e., a field equipped with an endomorphism) are equivalent to profinite groups equipped with an expansive endomorphism and a certain compatible difference Galois action. In any case, understanding the structure of expansive endomorphisms of profinite groups seems a worthwhile endeavor and that's what this talk is about. | |||||
17/02/2023 | 11.00 - 12.00 | V 205 / Zoom | Nathan Brownlowe | $C^{\ast}$-algebraic approaches to self-similarity | |
In this talk I will go through the basics of self-similar actions and some of their generalisations. I will then introduce $C^{\ast}$-algebras, before surveying the literature on how we build $C^{\ast}$-algebras to model self-similarity. | |||||
13.30 - 14.30 | V 205 / Zoom | Dave Robertson | Self-similar quantum groups | ||
Quantum automorphism groups originated in the work of Wang in the mid 90s as an answer to a question of Connes: what are the quantum automorphisms of a space? Wang showed that for a finite set with at least 4 points there are an infinite number of quantum permutations. Since then, work on quantum automorphism groups has progressed in many different directions, including the construction of the quantum automorphism group of a finite graph by Bichon in 2004 and quantum automorphisms of locally finite graphs by Rollier and Vaes in 2022. In a recent preprint with Nathan Brownlowe, we have shown that the quantum automorphism group of a homogeneous rooted tree is a compact quantum group, and defined when a quantum subgroup is self-similar. In this talk I will give an overview of this construction, and construct a number of examples through an analogue of the notion of a finitely constrained self-similar group defined by Sunic in 2011. | |||||
15.00 - 16.00 | V 205 / Zoom | Aidan Sims | $K$-theoretic duality for self-similar groupoids | ||
A $K$-theoretic duality for $C^{\ast}$-algebras is, roughly speaking, a particularly nice isomorphism of the $K$-theory groups of each with the $K$-homology groups of the other. They are generalisations of Poincare duality for manifolds, and in that vein, they often help to compute algebraic or analytic $K$-theory invariants in terms of more tractable topological information. Under some technical hypotheses, Nekrashevych established a $K$-theoretic duality between the $C^{\ast}$-algebra of a self-similar group and a related $C^{\ast}$-algebra associated to a limit space that resembles the way that real numbers are represented by decimal expansions. I will discuss how Nekrashevych’s limit space is constructed, focussing on elementary but instructive examples to keep things concrete, and sketch out how to use it to describe a $K$-theoretic duality that helps in computing $K$-theory for self-similar groupoid $C^{\ast}$-algebras. I won’t assume any background in any of this stuff. This is joint work with Brownlowe, Buss, Goncalves, Hume and Whittaker. | |||||
25/01/2023 | 15.00 - 16.00 | SR 118 / Zoom | Armin Weiss | An Automaton Group with PSPACE-Complete Word Problem | |
Finite automata pose an interesting alternative way to present groups and
semigroups. Some of these automaton groups became famous for their peculiar
properties and have been extensively studied. One aspect of this research is the study of algorithmic properties of automaton groups and semigroups. While many natural algorithmic decision problems have been proven or are generally suspected to be undecidable for these classes, the word problem forms a notable exception. In the group case, it asks whether a given word in the generators is equal to the neutral element in the group in question and is well-known to be decidable for automaton groups. In fact, it was observed in a work by Steinberg published in 2015 that it can be solved in nondeterministic linear space using a straight-forward guess and check algorithm. In the same work, he conjectured that there is an automaton group with a PSPACE-complete word problem. In a recent paper presented at STACS 2020, Jan Philipp Wächter and myself could prove that there indeed is such an automaton group. To achieve this, we combined two ideas. The first one is a construction introduced by D'Angeli, Rodaro and Wächter to show that there is an inverse automaton semigroup with a PSPACE-complete word problem and the second one is an idea already used by Barrington in 1989 to encode $\mathrm{NC}^{1}$ circuits in the group of even permutation over five elements. In the talk, we will discuss how Barrington's idea can be applied in the context of automaton groups, which will allow us to prove that the uniform word problem for automaton groups (were the generating automaton and, thus, the group is part of the input) is PSPACE- complete. Afterwards, we will also discuss the ideas underlying the construction to simulate a PSPACE- machine with an invertible automaton, which allow for extending the result to the non-uniform case. Finally, we will briefly look at related problems such as the compressed word problem for automaton groups and the special case of automaton group of polynomial activity. |
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25/11/2022 | 10.00 - 11.00 | SR118 / Zoom | Matthew Conder | Two-generator subgroups of $\mathrm{SL}_{2}$ over local fields | |
In this talk, we will give an overview of some results and open problems relating to two-generator subgroups of $\mathrm{SL}_2$ over a local field $K$. We first consider the archimedean setting, where certain discrete and/or free two-generator subgroups of $\mathrm{SL}(2,R)$ and $\mathrm{SL}(2,C)$ can be identified by investigating their respective actions by Möbius transformations on the upper half plane and Riemann sphere. We then outline some recent results in the non-archimedean setting, obtained by studying the analogous action of $\mathrm{SL}(2,K)$ by isometries on the corresponding Bruhat-Tits tree. Finally, we discuss an application of this work to the problem of deciding whether a two-generator subgroup of $\mathrm{SL}(2,K)$ is dense. | |||||
02/11/2022 | 14.00 - 15.00 | SR 202 / Zoom | Alexander Hulpke | Constructing Perfect Groups | |
The construction of perfect groups of a given order can be considered as the prototype of the construction of nonsolvable groups of a given order. I will describe a recent project to enumerate, up to isomorphism, the perfect groups of order up to $2\cdot 10^6$. It crucially relies on new tools for calculating cohomology, as well as improved implementations for isomorphism test. This work extends results of Holt and Plesken from 1989 and illustrates the scope of algorithmic improvements over the past decades. | |||||
07/10/2022 | 12.00 - 13.00 | V 205 / Zoom | Kane Townsend | Hyperbolic groups with $k$-geodetic Cayley graphs | |
A locally-finite simple connected graph is said to be $k$-geodetic for some $k\geq1$, if there is at most $k$ distinct geodesics between any two vertices of the graph. We investigate the properties of hyperbolic groups with $k$-geodetic Cayley graphs. To begin, we show that $k$-geodetic graphs cannot have a "ladder-like" geodesic structure with unbounded length. Using this bound, we generalise a well-known result of Papasoglu that states hyperbolic groups with $1$-geodetic Cayley graphs are virtually-free. We then investigate which elements of the hyperbolic group with $k$-geodetic Cayley graph commute with a given infinite order element. | |||||
14.30 - 15.30 | V 205 / Zoom | Eric Freden | Aspects of growth in Baumslag-Solitar groups | ||
In 1997, Grigorchuk and de la Harpe suggested computing the growth series for the Baumslag-Solitar group $\mathrm{BS}(2,3)$. After 25 years, this is still an open problem. In fact, the growth of only the solvable groups $\mathrm{BS}(1,n)$ and automatic groups $\mathrm{BS}(n,n)$ are known. In this talk I will review what has since been discovered about these remarkable groups and conclude with new unpublished results concerning the exponents of growth for the subfamily $\mathrm{BS}(2,2n)$. | |||||
16.00 - 17.00 | V 205 / Zoom | Volker Diekert | Decidability of membership problems for $2\times 2$ matrices over $\mathbb{Q}$ | ||
My talk is based on a joint work with Igor Potapov and Pavel Semukhin (Liverpool, UK). We consider membership problems in matrix semigroups. Using symbolic algorithms on words and finite automata, we prove various new decidability results for $2\times 2$ matrices over $\mathbb{Q}$. For that, we introduce the concept of flat rational sets: if $M$ is a monoid and $N$ is a submonoid, then flat rational sets of $M$ over $N$ are finite unions of the form $L_0g_1L_1 \cdots g_t L_t$ where all $L_i$'s are rational subsets of $N$ and $g_i\in M$. We give quite general sufficient conditions under which flat rational sets form an effective relative Boolean algebra. As a corollary, we obtain that the emptiness problem for Boolean combinations of flat rational subsets of $\mathrm{GL}(2,\mathbb{Q})$ over $\mathrm{GL}(2,\mathbb{Z})$ is decidable (in singly exponential time). It is possible that such a strong decidability result cannot be pushed any further for groups sitting between $\mathrm{GL}(2,\mathbb{Z})$ and $\mathrm{GL}(2,\mathbb{Q})$. We also show a dichotomy for nontrivial group extension of $\mathrm{GL}(2,\mathbb{Z})$ in $\mathrm{GL}(2,\mathbb{Q})$: if $G$ is a f.g. group such that $\mathrm{GL}(2,\mathbb{Z}) < G \leq \mathrm{GL}(2,\mathbb{Q})$, then either $G\cong \mathrm{GL}(2,\mathbb{Z})\times \mathbb{Z}^k$, for some $k\geq 1$, or $G$ contains an extension of the Baumslag-Solitar group $\mathrm{BS}(1,q)$, with $q\geq 2$, of infinite index. In the first case of the dichotomy the membership problem for $G$ is decidable but the equality problem for rational subsets of $G$ is undecidable. In the second case, decidability of the membership problem for rational subsets in $G$ is open. Our result improves various natural decidability results for $2 \times 2$ matrices with rational entries, and it also supports them with concrete complexity bounds for the first time. |
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01/07/2022 | 10.00 - 11.00 | W 202 / Zoom | Claudio Bravo | Quotients of the Bruhat-Tits tree function field analogs of the Hecke congruence subgroups | |
Let $C$ be a smooth, projective, and geometrically connected curve defined over a finite field $F$. For each closed point $P_\infty$ of $C$, let $R$ be the ring of functions that are regular outside $P_\infty$, and let $K$ be the completion path $P_\infty$ of the function field of $C$. In order to study group of the form $\mathrm{GL}_2(R)$, Serre describes the quotient graph $\mathrm{GL}_2(R)\backslash T$, where $T$ is the Bruhat-Tits tree defined from $\mathrm{SL}_2(K)$. In particular, Serre shows that $\mathrm{GL}_2(R)\backslash T$ is the union of a finite graph and a finite number of ray shaped subgraphs, which are called cusps. It is not hard to see that finite index subgroups inherit this property. In this exposition we describe the quotient graph $H\backslash T$ defined from the action on $T$ of the group $H$ consisting of matrices that are upper triangular modulo $I$, where $I$ is an ideal $R$. More specifically, we give an explicit formula for the cusp number $H\backslash T$. Then By, using Bass-Serre theory, we describe the combinatorial structure of $H$. These groups play, in the function field context, the same role as the Hecke Congruence subgroups of $\mathrm{SL}_2(Z)$. Moreover, not that the groups studied by Serre correspond to the case where the ideal $I$ coincides with the ring $R$. | |||||
11.30 - 12.30 | W 202 / Zoom | Matthew Conder | Discrete two-generator subgroups of $\mathrm{PSL}(2,\mathbb{Q}_p)$ | ||
Due to work of Gilman, Rosenberger, Purzitsky and many others, discrete two-generator subgroups of $\mathrm{PSL}(2,\mathbb{R})$ have been completely classified by studying their action by Möbius transformations on the hyperbolic plane. Here we aim to classify discrete two-generator subgroups of $\mathrm{PSL}(2,\mathbb{Q}_p)$ by studying their action by isometries on the Bruhat-Tits tree. We first give a general structure theorem for two-generator groups acting by isometries on a tree, which relies on certain Klein-Maskit combination theorems. We will then discuss how this theorem can be applied to determine discreteness of a two-generator subgroup of $\mathrm{PSL}(2,\mathbb{Q}_p)$. This is ongoing work in collaboration with Jeroen Schillewaert. | |||||
14.00 - 15.00 | W 202 / Zoom | George Willis | Groups acting on regular trees and t.d.l.c. groups | ||
Groups of automorphisms of regular trees are an important source of examples of and intuition about totally disconnected, locally compact (t.d.l.c.) groups. Indeed, Pierre-Emmanuel Caprice has called them a microcosm the general theory of t.d.l.c. groups. Although much is know about them, many questions remain open. This talk will survey some of what is known about groups of tree automorphisms and how it relates to the general theory. | |||||
01/04/2022 | 12.00 - 13.00 | US 321 / Zoom | Max Carter | Unitary representations and the type I property of groups acting on trees | |
Unitary representations are a classical and useful tool for studying locally compact groups: motivated in part by quantum mechanics, they have been studied in detail since the early-mid 1900’s with much success, and they enable group theorists to employ functional analytic techniques in the study of locally compact groups. The algebras that unitary representations generate play an important role in not only understanding the representation theory of a locally compact group, but also in understanding properties pertaining to the group itself. This talk will give a brief introduction to some of the basics of the unitary representation theory of locally compact groups, with focus placed on the associated operator algebraic structures/properties. In particular, 'type I groups' and 'CCR groups' will be the main focus. As an application, I will discuss some current research interests in the unitary representation theory of groups acting on trees, including work of myself on the unitary representation theory of `scale groups’. | |||||
14.30 - 15.30 | US 321 / Zoom | Camila Sehnem | Equilibrium on Toeplitz extensions of higher dimensional noncommutative tori | ||
The $C^*$-algebra generated by the left-regular representation of $\mathbb{N}^n$ twisted by a $2$-cocycle is a Toeplitz extension of an $n$-dimensional noncommutative torus, on which each vector $r \in [0,\infty)^n$ determines a one-parameter subgroup of the gauge action. I will report on joint work with Z. Afsar, J. Ramagge and M. Laca, in which we show that the equilibrium states of the resulting C*-dynamical system are parametrised by tracial states of the noncommutative torus corresponding to the restriction of the cocycle to the vanishing coordinates of $r$. These in turn correspond to probability measures on a classical torus whose dimension depends on a certain degeneracy index of the restricted cocycle. Our results generalise the phase transition on the Toeplitz noncommutative tori used as building blocks in work of Brownlowe, Hawkins and Sims, and of Afsar, an Huef, Raeburn and Sims. | |||||
16.00 - 17.00 | US 321 / Zoom | Roozbeh Hazrat | Sandpile models and Leavitt algebras | ||
Sandpile models are about how things spread along a grid (think of Covid!) and Leavitt algebras are algebras associated to graphs. We relate these two subjects! | |||||
04/03/2022 | 12.00 - 13.00 | US 321 / Zoom | Michal Ferov | Automorphism groups of Cayley graphs of Coxeter groups: when are they discrete? | |
Group of automorphisms of a connected locally finite graph is naturally a totally disconnected locally compact topological group, when equipped with the permutation topology. It therefore makes sense to ask for which graphs is the topology not discrete. We show that in case of Cayley graphs of Coxeter groups, one can fully characterise the discrete ones in terms of the symmetries of the corresponding Coxeter system. Joint work with Federico Berlai. | |||||
14.30 - 15.30 | US 321 / Zoom | Jeroen Schillewaert | The geometries of the Freudenthal-Tits magic square | ||
I will give an overview of a programme investigating projective embeddings of (exceptional) geometries which Hendrik Van Maldeghem and I started in 2010. | |||||
16.00 - 17.00 | US 321 / Zoom | James Parkinson | Automorphisms and opposition in spherical buildings | ||
The geometry of elements fixed by an automorphism of a spherical building is a rich and well-studied object, intimately connected to the theory of Galois descent in buildings. In recent years, a complementary theory has emerged investigating the geometry of elements mapped onto opposite elements by a given automorphism. In this talk we will give an overview of this theory. This work is joint primarily with Hendrik Van Maldeghem (along with others). | |||||
04/02/2022 | 12.00 - 13.00 | SR 118 / Zoom | Anne Thomas | A gallery model for affine flag varieties via chimney retractions | |
We provide a unified combinatorial framework to study orbits in affine flag varieties via the associated Bruhat-Tits buildings. We first formulate, for arbitrary affine buildings, the notion of a chimney retraction. This simultaneously generalises the two well-known notions of retractions in affine buildings: retractions from chambers at infinity and retractions from alcoves. We then present a recursive formula for computing the images of certain minimal galleries in the building under chimney retractions, using purely combinatorial tools associated to the underlying affine Weyl group. Finally, for Bruhat-Tits buildings, we relate these retractions and their effect on certain minimal galleries to double coset intersections in the corresponding affine flag variety. This is joint work with Elizabeth Milicevic, Yusra Naqvi and Petra Schwer. | |||||
14.30 - 15.30 | SR 118 / Zoom | Sebastian Bischof | (Twin) Buildings and groups | ||
Buildings have been introduced by Tits in order to study semi-simple algebraic groups from a geometrical point of view. One of the most important results in the theory of buildings is the classification of thick irreducible spherical buildings of rank at least 3. In particular, any such building comes from an RGD-system. The decisive tool in this classification is the Extension theorem for spherical buildings, i.e. a local isometry extends to the whole building. Twin buildings were introduced by Ronan and Tits in the late 1980s. Their definition was motivated by the theory of Kac-Moody groups over fields. Each such group acts naturally on a pair of buildings and the action preserves an opposition relation between the chambers of the two buildings. This opposition relation shares many important properties with the opposition relation on the chambers of a spherical building. Thus, twin buildings appear to be natural generalizations of spherical buildings with infinite Weyl group. Since the notion of RGD-systems exists not only in the spherical case, one can ask whether any twin building (satisfying some further conditions) comes from an RGD-system. In 1992 Tits proves several results that are inspired by his strategy in the spherical case and he discusses several obstacles for obtaining a similar Extension theorem for twin buildings. In this talk I will speak about the history and developments of the Extension theorem for twin buildings. |
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15/11/2021 | 15.30 - 16.30 | Zoom | Jingyin Huang | The Helly geometry of some Garside and Artin groups | |
Artin groups emerged from the study of braid groups and complex hyperplane arrangements. Artin groups have very simple presentation, yet rather mysterious geometry with many basic questions widely open. I will present a way of understanding certain Artin groups and Garside groups by building geometric models on which they act. These geometric models are non-positively curved in an appropriate sense, and such curvature structure yields several new results on the algorithmic, topological and geometric aspects of these groups. No previous knowledge on Artin groups or Garside groups is required. This is joint work with D. Osajda. | |||||
17.00 - 18.00 | Zoom | Suraj Krishna | The mapping torus of a torsion-free hyperbolic group is relatively hyperbolic | ||
Let $G$ be the fundamental group of a closed orientable surface of genus at least 2, and $\alpha$ an automorphism of $G$. In a celebrated result, Thurston showed that the mapping torus $G \rtimes_{\alpha} \mathbb{Z}$ is hyperbolic if and only if no power of $\alpha$ preserves a non-trivial conjugacy class. In this talk, I will describe joint work with François Dahmani, where we show that if $G$ is torsion-free hyperbolic, then $G\rtimes_{\alpha} \mathbb{Z}$ is relatively hyperbolic with ``optimal'' parabolic subgroups. | |||||
01/11/2021 | 17.30 - 18.30 | Zoom | Sanghyun Kim | Optimal regularity of mapping class group actions on the circle | |
We prove that for each finite index subgroup $H$ of the mapping class group of a closed hyperbolic surface, and for each real number $r>0$ there does not exist a faithful $C^{1+r}$-action of $H$ on a circle. (Joint with Thomas Koberda and Cristobal Rivas) | |||||
19.00 - 20.00 | Zoom | Francois Thilmany | Uniform discreteness of arithmetic groups and the Lehmer conjecture | ||
The famous Lehmer problem asks whether there is a gap between 1 and the Mahler measure of algebraic integers which are not roots of unity. Asked in 1933, this deep question concerning number theory has since then been connected to several other subjects. After introducing the concepts involved, we will briefly describe a few of these connections with the theory of linear groups. Then, we will discuss the equivalence of a weak form of the Lehmer conjecture and the "uniform discreteness" of cocompact lattices in semisimple Lie groups (conjectured by Margulis). Joint work with Lam Pham. | |||||
27/09/2021 | 16.30 - 17.30 | Zoom | Siegfried Echterhoff | Amenable group actions on $C^{\ast}$-algebras and the weak containment problem | |
The notion of amenable actions by discrete groups on $C^{\ast}$-algebras has been introduced by Claire Amantharaman-Delaroche more than thirty years ago, and has become a well understood theory with many applications. So it is somewhat surprising that an established theory of amenable actions by general locally compact groups has been missed for a very long time. We now present a theory which extends the discrete case and unifies several notions of approximation properties of actions which have been discussed in the literature. We also discuss the weak containment problem which asks wether an action $\alpha:G\to \mathrm{Aut}(A)$ is amenable if and only if the maximal and reduced crossed products coincide.
In this lecture we report on joint work with Alcides Buss and Rufus Willett. |
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18.00 - 19.00 | Zoom | Tim de Laat | Gelfand pairs, spherical functions and exotic group $C^{\ast}$-algebras | ||
For a non-amenable group $G$, there can be many group $C^{\ast}$-algebras that lie naturally between the universal and the reduced $C^{\ast}$-algebra of $G$. These are called exotic group $C^{\ast}$-algebras. After a short introduction, I will explain that if $G$ is a simple Lie group or an appropriate locally compact group acting on a tree, the $L^p$-integrability properties of different spherical functions on $G$ (relative to a maximal compact subgroup) can be used to distinguish between exotic group $C^{\ast}$-algebras. This recovers results of Samei and Wiersma. Additionally, I will explain that under certain natural assumptions, the aforementioned exotic group $C^{\ast}$-algebras are the only ones coming from $G$-invariant ideals in the Fourier-Stieltjes algebra of $G$.
This is based on joint work with Dennis Heinig and Timo Siebenand. |
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30/08/2021 | 18.00 - 19.00 | Zoom | George Willis | Constructing groups with flat-rank greater than $1$ | |
The contraction subgroup for $x$ in the locally compact group, $G$, $\mathrm{con}(x) = \left\{ g\in G \mid x^ngx^{-n} \to 1\text{ as }n\to\infty \right\}$, and the Levi subgroup is $\mathrm{lev}(x) = \left\{ g\in G \mid \{x^ngx^{-n}\}_{n\in\mathbb{Z}} \text{ has compact closure}\right\}$. The following will be shown.
Let $G$ be a totally disconnected, locally compact group and $x\in G$. Let $y\in{\sf lev}(x)$. Then there are $x'\in G$ and a compact subgroup, $K\leq G$ such that: -$K$ is normalised by $x'$ and $y$, -$\mathrm{con}(x') = \mathrm{con}(x)$ and $\mathrm{lev}(x') = \mathrm{lev}(x)$ and -the group $\langle x',y,K\rangle$ is abelian modulo $K$, and hence flat. If no compact open subgroup of $G$ normalised by $x$ and no compact open subgroup of $\mathrm{lev}(x)$ normalised by $y$, then the flat-rank of $\langle x',y,K\rangle$ is equal to $2$. |
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09/08/2021 | 16.30 - 17.30 | Zoom | Sven Raum | Locally compact groups acting on trees, the type I conjecture and non-amenable von Neumann algebras | |
In the 90's, Nebbia conjectured that a group of tree automorphisms acting transitively on the tree's boundary must be of type I, that is, its unitary representations can in principal be classified. For key examples, such as Burger-Mozes groups, this conjecture is verified. Aiming for a better understanding of Nebbia's conjecture and a better understanding of representation theory of groups acting on trees, it is natural to ask whether there is a characterisation of type I groups acting on trees. In 2016, we introduced in collaboration with Cyril Houdayer a refinement of Nebbia's conjecture to a trichotomy, opposing type I groups with groups whose von Neumann algebra is non-amenable. For large classes of groups, including Burger-Mozes groups, we could verify this trichotomy.
In this talk, I will motivate and introduce the conjecture trichotomy for groups acting on tress and explain how von Neumann algebraic techniques enter the picture. |
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18.00 - 19.00 | Zoom | James Parkinson | Automata for Coxeter groups | ||
In 1993 Brink and Howlett proved that finitely generated Coxeter groups are automatic. In particular, they constructed a finite state automaton recognising the language of reduced words in the Coxeter group. This automaton is constructed in terms of the remarkable set of "elementary roots" in the associated root system.
In this talk we outline the construction of Brink and Howlett. We also describe the minimal automaton recognising the language of reduced words, and prove necessary and sufficient conditions for the Brink-Howlett automaton to coincide with this minimal automaton. This resolves a conjecture of Hohlweg, Nadeau, and Williams, and is joint work with Yeeka Yau. |
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05/07/2021 | 18.00 - 19.00 | Zoom | Lancelot Semal | Unitary representations of totally disconnected locally compact groups satisfying Ol'shanskii's factorization | |
We provide a new axiomatic framework, inspired by the work of Ol'shanskii, to describe explicitly certain irreducible unitary representations of second-countable non-discrete unimodular totally disconnected locally compact groups. We show that this setup applies to various families of automorphism groups of locally finite semiregular trees and right-angled buildings.
The talk is based on material presented in arxiv.org/abs/2106.05730. |
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21/06/2021 | 16.30 - 17.30 | Zoom | Yves Stalder | Highly transitive groups among groups acting on trees | |
Highly transitive groups, i.e. groups admitting an embedding in $\mathrm{Sym}(\mathbb{N})$ with dense image, form a wide class of groups. For instance, M. Hull and D. Osin proved that it contains all countable acylindrically hyperbolic groups with trivial finite radical. After an introduction to high transitiviy, I will present a theorem (from joint work with P. Fima, F. Le Maître and S. Moon) showing that many groups acting on trees are highly transitive. On the one hand, this theorem gives new examples of highly transitive groups. On the other hand, it is sharp because of results by A. Le Boudec and N. Matte Bon. | |||||
18.00 - 19.00 | Zoom | Ilaria Castellano | The Euler characteristic and the zeta-functions of a totally disconnected locally compact group | ||
The Euler-Poincaré characteristic of a discrete group is an important (but also quite mysterious) invariant. It is usually just an integer or a rational number and reflects many quite significant properties. The realm of totally disconnected locally compact groups admits an analogue of the Euler-Poincaré characteristic which surprisingly is no longer just an integer, or a rational number, but a rational multiple of a Haar measure. Warning: in order to gain such an invariant the group has to be unimodular and satisfy some cohomological finiteness conditions. Examples of groups satisfying these additional conditions are the fundamental groups of finite trees of profinite groups. What arouses our curiosity is the fact that - in some cases - the Euler-Poincaré characteristic turns out to be miraculously related to a zeta-function. A large part of the talk will be devoted to the introduction of the just-cited objects. We aim at concluding the presentation by facing the concrete example of the group of $F$-points of a split semisimple simply connected algebraic group $G$ over $F$ (where $F$ denotes a non-archimedean locally compact field of residue characteristic $p$). Joint work with Gianmarco Chinello and Thomas Weigel. | |||||
07/06/2021 | 16.30 - 17.30 | Zoom | Waldemar Hołubowski | Normal subgroups in the group of column-finite infinite matrices | |
The classical result, due to Jordan, Burnside, Dickson, says that every normal subgroup of $\mathrm{GL}(n, K)$ ($K$ - a field, $n \geq 3$) which is not contained in the center, contains $\mathrm{SL}(n, K)$. A. Rosenberg gave description of normal subgroups of $\mathrm{GL}(V)$, where $V$ is a vector space of any infinite cardinality dimension over a division ring. However, when he considers subgroups of the direct product of the center and the group of linear transformations $g$ such that $g-id_V$ has finite dimensional range the proof is not complete. We fill this gap for countably dimensional $V$ giving description of the lattice of normal subgroups in the group of infinite column-finite matrices indexed by positive integers over any field. Similar results for Lie algebras of matrices will be surveyed.
The is based on results presented in https://arxiv.org/abs/1808.06873 and https://arxiv.org/abs/1806.01099. (joint work with Martyna Maciaszczyk and Sebastian Zurek.) |
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24/05/2021 | 16.30 - 17.30 | Zoom | Libor Barto | CSPs and Symmetries | |
How difficult is it to solve a given computational problem? In a large class of computational problems, including the fixed-template Constraint Satisfaction Problems (CSPs), this fundamental question has a simple and beautiful answer: the more symmetrical the problem is, the easier is to solve it. The tight connection between the complexity of a CSP and a certain concept that captures its symmetry has fueled much of the progress in the area in the last 20 years. I will talk about this connection and some of the many tools that have been used to analyze the symmetries. The tools involve rather diverse areas of mathematics including algebra, analysis, combinatorics, logic, probability, and topology. | |||||
18.00 - 19.00 | Zoom | Zoé Chatzidakis | A new invariant for difference fields | ||
If $(K,f)$ is a difference field, and $a$ is a finite tuple in some difference field extending $K$, and such that $f(a)$ in $K(a)^{\mathrm{alg}}$, then we define $dd(a/K)=lim[K(f^k(a),a):K(a)]^{1/k}$, the distant degree of $a$ over $K$. This is an invariant of the difference field extension $K(a)^{\mathrm{alg}}/K$. We show that there is some $b$ in the difference field generated by $a$ over $K$, which is equi-algebraic with $a$ over $K$, and such that $dd(a/K)=[K(f(b),b):K(b)]$, i.e.: for every $k>0$, $f(b)$ in $K(b,f^k(b))$. Viewing $Aut(K(a)^{\mathrm{alg}}/K)$ as a locally compact group, this result is connected to results of Goerge Willis on scales of automorphisms of locally compact totally disconnected groups. I will explicit the correspondence between the two sets of results. (Joint with E. Hrushovski) | |||||
10/05/2021 | 16.30 - 17.30 | Zoom | Yago Antolin | Geometry and Complexity of positive cones in groups | |
A positive cone on a group $G$ is a subsemigroup $P$, such that $G$ is the disjoint union of $P$, $P^{-1}$ and the trivial element. Positive cones codify naturally $G$-left-invariant total orders on $G$. When $G$ is a finitely generated group, we will discuss whether or not a positive cone can be described by a regular language over the generators and how the ambient geometry of $G$ influences the geometry of a positive cone. This will be based on joint works with Juan Alonso, Joaquin Brum, Cristobal Rivas and Hang Lu Su. | |||||
18.00 - 19.00 | Zoom | Robert Kropholler | Groups of type $FP_2$ over fields but not over the integers | ||
Being of type $FP_2$ is an algebraic shadow of being finitely presented. A long standing question was whether these two classes are equivalent. This was shown to be false in the work of Bestvina and Brady. More recently, there are many new examples of groups of type $FP_2$ coming with various interesting properties. I will begin with an introduction to the finiteness property $FP_2$. I will end by giving a construction to find groups that are of type $FP_2(F)$ for all fields $F$ but not $FP_2(Z)$. | |||||
19/04/2021 | 18.00 - 19.00 | Zoom | Laura Ciobanu | Free group homomorphisms and the Post Correspondence Problem | |
The Post Correspondence Problem (PCP) is a classical problem in computer science that can be stated as: is it decidable whether given two morphisms $g$ and $h$ between two free semigroups $A$ and $B$, there is any nontrivial $x$ in $A$ such that $g(x)=h(x)$? This question can be phrased in terms of equalisers, asked in the context of free groups, and expanded: if the `equaliser' of $g$ and $h$ is defined to be the subgroup consisting of all $x$ where $g(x)=h(x)$, it is natural to wonder not only whether the equaliser is trivial, but what its rank or basis might be. While the PCP for semigroups is famously insoluble and acts as a source of undecidability in many areas of computer science, the PCP for free groups is open, as are the related questions about rank, basis, or further generalisations. However, in this talk we will show that there are links and surprising equivalences between these problems in free groups, and classes of maps for which we can give complete answers. This is joint work with Alan Logan. | |||||
08/03/2021 | 18.30 - 19.30 | Zoom | Charlotte Hoffmann | Short words of high imprimitivity rank yield hyperbolic one-relator groups | |
It is a long standing question whether a group of type $F$ that does not contain Baumslag–Solitar subgroups is necessarily hyperbolic. One-relator groups are of type $F$ and Louder and Wilton showed that if the defining relator has imprimitivity rank greater than $2$, they do not contain Baumslag-Solitar subgroups, so they conjecture that such groups are hyperbolic. Cashen and I verified the conjecture computationally for relators of length at most $17$. In this talk I'll introduce hyperbolic groups and the imprimitivity rank of elements in a free group. I’ll also discuss how to verify hyperbolicity using versions of combinatorial curvature on van Kampen diagrams. | |||||
19.30 - 21.00 | Zoom | Dawid Kielak | Recognising surface groups | ||
I will address two problems about recognising surface groups. The first one is the classical problem of classifying Poincaré duality groups in dimension two. I will present a new approach to this, joint with Peter Kropholler. The second problem is about recognising surface groups among one-relator groups. Here I will present a new partial result, joint with Giles Gardam and Alan Logan. | |||||
22/02/2021 | 18.30 - 19.30 | Zoom | Paul-Henry Leemann | Cayley graphs with few automorphisms | |
Let $G$ be a group and $S$ a generating set. Then the group $G$ naturally acts on the Cayley graph $\mathrm{Cay}(G,S)$ by left multiplications. The group $G$ is said to be rigid if there exists an $S$ such that the only automorphisms of $\mathrm{Cay}(G,S)$ are the ones coming from the action of $G$. While the classification of finite rigid groups was achieved in 1981, few results were known about infinite groups. In a recent work, with M. de la Salle we gave a complete classification of infinite finitely generated rigid groups. As a consequence, we also obtain that every finitely generated group admits a Cayley graph with countable automorphism group. | |||||
20.00 - 21.00 | Zoom | Giles Gardam | Kaplansky's conjectures (Slides) | ||
Kaplansky made various related conjectures about group rings, especially for torsion-free groups. For example, the zero divisors conjecture predicts that if $K$ is a field and $G$ is a torsion-free group, then the group ring $K[G]$ has no zero divisors. I will survey what is known about the conjectures, including their relationships to each other and to other group properties such as orderability, and present some recent progress. | |||||
25/01/2021 | 18.30 - 19.30 | Zoom | François Le Maître | Dense totipotent free subgroups of full groups | |
In this talk, we will be interested in measure-preserving actions of countable groups on standard probability spaces, and more precisely in the partitions of the space into orbits that they induce, also called measure-preserving equivalence relations. In 2000, Gaboriau obtained a characterization of the ergodic equivalence relations which come from non-free actions of the free group on $n>1$ generators: these are exactly the equivalence relations of cost less than n. A natural question is: how non-free can these actions be made, and what does the action on each orbit look like? We will obtain a satisfactory answer by showing that the action on each orbit can be made totipotent, which roughly means "as rich as possible", and furthermore that the free group can be made dense in the ambient full group of the equivalence relation. This is joint work with Alessandro Carderi and Damien Gaboriau. | |||||
20.00 - 21.00 | Zoom | Charles Cox | Spread and infinite groups | ||
My recent work has involved taking questions asked for finite groups and considering them for infinite groups. There are various natural directions with this. In finite group theory, there exist many beautiful results regarding generation properties. One such notion is that of spread, and Scott Harper and Casey Donoven have raised several intriguing questions for spread for infinite groups (in https://arxiv.org/abs/1907.05498). A group $G$ has spread $k$ if for every $g_1,\ldots,g_k$ we can find an $h$ in $G$ such that $\langle g_i, h\rangle=G$. For any group we can say that if it has a proper quotient that is non-cyclic, then it has spread $0$. In the finite world there is then the astounding result - which is the work of many authors - that this condition on proper quotients is not just a necessary condition for positive spread, but is also a sufficient one. Harper-Donoven’s first question is therefore: is this the case for infinite groups? Well, no. But that’s for the trivial reason that we have infinite simple groups that are not 2-generated (and they point out that 3-generated examples are also known). But if we restrict ourselves to 2-generated groups, what happens? In this talk we’ll see the answer to this question. The arguments will be concrete (*) and accessible to a general audience. (*) at the risk of ruining the punchline, we will find a 2-generated group that has every proper quotient cyclic but that has spread zero. |
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23/11/2020 | 18.30 - 19.30 | Zoom | William Hautekiet | Automorphism groups of transcendental field extensions | |
It is well-known that the Galois group of an (infinite) algebraic field extension is a profinite group. When the extension is transcendental, the automorphism group is no longer compact, but has a totally disconnected locally compact structure (TDLC for short). The study of TDLC groups was initiated by van Dantzig in 1936 and then restarted by Willis in 1994. In this talk some of Willis' concepts, such as tidy subgroups, the scale function, flat subgroups and directions are introduced and applied to examples of automorphism groups of transcendental field extensions. It remains unknown whether there exist conditions that a TDLC group must satisfy to be a Galois group. A suggestion of such a condition is made. | |||||
20.00 - 21.00 | Zoom | Florian Breuer | Realising general linear groups as Galois groups | ||
I will show how to construct field extensions with Galois groups isomorphic to general linear groups (with entries in various rings and fields) from the torsion of elliptic curves and Drinfeld modules. No prior knowledge of these structures is assumed. | |||||
09/11/2020 | 20.00 - 21.00 | Zoom | Henry Bradford | Quantitative LEF and topological full groups | |
Topological full groups of minimal subshifts are an important source of exotic examples in geometric group theory, as well as being powerful invariants of symbolic dynamical systems. In 2011, Grigorchuk and Medynets proved that TFGs are LEF, that is, every finite subset of the multiplication table occurs in the multiplication table of some finite group. In this talk we explore some ways in which asymptotic properties of the finite groups which occur reflect asymptotic properties of the associated subshift. Joint work with Daniele Dona. | |||||
16/10/2020 | 10.00 - 11.00 | Zoom | Rachel Skipper | Maximal Subgroups of Thompson's group V | |
There has been a long interest in embedding and non-embedding results for groups in the Thompson family. One way to get at results of this form is to classify maximal subgroups. In this talk, we will define certain labelings of binary trees and use them to produce a large family of new maximal subgroups of Thompson's group V. We also relate them to a conjecture about Thompson's group T. This is joint, ongoing work with Jim Belk, Collin Bleak, and Martyn Quick at the University of Saint Andrews. |
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11.30 - 12.30 | Zoom | Lawrence Reeves | Irrational-slope versions of Thompson’s groups T and V | ||
We consider irrational slope versions of T and V. We give infinite presentations for these groups and show how they can be represented by tree-pair diagrams. We also show that they have index-2 normal subgroups that are simple. This is joint work with Brita Nucinkis and Pep Burillo. |
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02/10/2020 | 16.00 - 17.00 | Zoom | Alejandra Garrido | When is a piecewise (a.k.a. topological) full group locally compact? | |
Answer: Only when it's an ample group in the sense of Krieger (in particular, discrete, countable and locally finite) and has a Bratteli diagram satisfying certain conditions. Complaint: Wait, isn't Neretin's group a non-discrete, locally compact, topological full group? Retort: It is, but you need to use the correct topology! A fleshed-out version of the above conversation will be given in the talk. Based on joint work with Colin Reid. |
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17.30 - 18.30 | Zoom | Feyisayo Olukoya | The group of automorphisms of the shift dynamical system and the Higman-Thompson groups | ||
We give a survey of recent results exploring connections between the Higman-Thompson groups and their automorphism groups and the group of autmorphisms of the shift dynamical system. Our survey takes us from dynamical systems to group theory via groups of homeomorphisms with a segue through combinatorics, in particular, de Bruijn graphs. Joint work with Collin Bleak and Peter Cameron. |
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18/09/2020 | 15.00 - 16.00 | Zoom | Gabriel Verret | Local actions in vertex-transitive graphs | |
A graph is vertex-transitive if its group of automorphism acts transitively on its vertices. A very important concept in the study of these graphs is that of local action, that is, the permutation group induced by a vertex-stabiliser on the corresponding neighbourhood. I will explain some of its importance and discuss some attempts to generalise it to the case of directed graphs. | |||||
16.30 - 17.30 | Zoom | Michael Giudici | The synchronisation hierarchy for permutation groups | ||
The concept of a synchronising permutation group was introduced nearly 15 years ago as a possible way of approaching The Černý Conjecture. Such groups must be primitive. In an attempt to understand synchronising groups, a whole hierarchy of properties for a permutation group has been developed, namely, 2-transitive groups, $\mathbb{Q}$I-groups, spreading, separating, synchronsing, almost synchronising and primitive. Many surprising connections with other areas of mathematics such as finite geometry, graph theory, and design theory have arisen in the study of these properties. In this survey talk I will give an overview of the hierarchy and discuss what is known about which groups lie where. | |||||
04/09/2020 | 15.00 - 16.00 | Zoom | Murray Elder | Rewriting systems and geodetic graphs (Slides) | |
I will describe a new proof, joint with Adam Piggott (UQ), that groups presented by finite convergent length-reducing rewriting systems where each rule has left-hand side of length 3 are exactly the plain groups (free products of finite and infinite cyclic groups). Our proof relies on a new result about properties of embedded circuits in geodetic graphs, which may be of independent interest in graph theory. | |||||
16.30 - 17.30 | Zoom | Ana Khukhro | A new characterisation of virtually free groups (Notes) | ||
A finite graph that can be obtained from a given graph by contracting edges and removing vertices and edges is said to be a minor of this graph. Minors have played an important role in graph theory, ever since the well-known result of Kuratowski that characterised planar graphs as those that do not admit the complete graph on 5 vertices nor the complete bipartite graph on (3,3) vertices as minors. In this talk, we will explore how this concept interacts with some notions from geometric group theory, and describe a new characterisation of virtually free groups in terms of minors of their Cayley graphs. | |||||
21/08/2020 | 15.00 - 16.00 | Zoom | Kasia Jankiewicz | Residual finiteness of certain 2-dimensional Artin groups | |
We show that many 2-dimensional Artin groups are residually finite. This includes Artin groups on three generators with labels at least 3, where either at least one label is even, or at most one label is equal 3. The result relies on decomposition of these Artin groups as graphs of finite rank free groups. | |||||
16.30 - 17.30 | Zoom | Simon Smith | Infinite primitive permutation groups, cartesian decompositions, and topologically simple locally compact groups (Slides) | ||
A non-compact, compactly generated, locally compact group whose proper quotients are all compact is called just-non-compact. Discrete just-non-compact groups are John Wilson’s famous just infinite groups. In this talk, I’ll describe an ongoing project to use permutation groups to better understand the class of just-non-compact groups that are totally disconnected. An important step for this project has recently been completed: there is now a structure theorem for non-compact tdlc groups G that have a compact open subgroup that is maximal. Using this structure theorem, together with Cheryl Praeger and Csaba Schneider’s recent work on homogeneous cartesian decompositions, one can deduce a neat test for whether the monolith of such a group G is a one-ended group in the class S of nondiscrete, topologically simple, compactly generated, tdlc groups. This class S plays a fundamental role in the structure theory of compactly generated tdlc groups, and few types of groups in S are known. | |||||
07/08/2020 | 15.00 - 16.00 | Zoom | Tony Guttmann | On the amenability of Thompson's Group $F$ | |
In 1967 Richard Thompson introduced the group $F$, hoping that it was non-amenable, since then it would disprove the von Neumann conjecture. Though the conjecture has subsequently been disproved, the question of the amenability of Thompson's group F has still not been rigorously settled. In this talk I will present the most comprehensive numerical attack on this problem that has yet been mounted. I will first give a history of the problem, including mention of the many incorrect "proofs" of amenability or non-amenability. Then I will give details of a new, efficient algorithm for obtaining terms of the co-growth sequence. Finally I will describe a number of numerical methods to analyse the co-growth sequences of a number of infinite, finitely-generated groups, and show how these methods provide compelling evidence (though of course not a proof) that Thompson's group F is not amenable. I will also describe an alternative route to a rigorous proof. (This is joint work with Andrew Elvey Price). | |||||
16.30 - 17.30 | Zoom | Collin Bleak | On the complexity of elementary amenable subgroups of R. Thompson's group $F$ | ||
The theory of EG, the class of elementary amenable groups, has developed steadily since the class was introduced constructively by Day in 1957. At that time, it was unclear whether or not EG was equal to the class AG of all amenable groups. Highlights of this development certainly include Chou's article in 1980 which develops much of the basic structure theory of the class EG, and Grigorchuk's 1985 result showing that the first Grigorchuk group $\Gamma$ is amenable but not elementary amenable. In this talk we report on work where we demonstrate the existence of a family of finitely generated subgroups of Richard Thompson’s group $F$ which is strictly well-ordered by the embeddability relation in type $\varepsilon_{0}+1$. All except the maximum element of this family (which is $F$ itself) are elementary amenable groups. In this way, for each $\alpha<\varepsilon_{0}$, we obtain a ﬁnitely generated elementary amenable subgroup of F whose EA-class is $\alpha+2$. The talk will be pitched for an algebraically inclined audience, but little background knowledge will be assumed. Joint work with Matthew Brin and Justin Moore. | |||||
17/07/2020 | 15.00 - 16.00 | Zoom | Harry Hyungryul Baik | Normal generators for mapping class groups are abundant in the fibered cone (Notes) | |
We show that for almost all primitive integral cohomology classes in the fibered cone of a closed fibered hyperbolic 3-manifold, the monodromy normally generates the mapping class group of the fiber. The key idea of the proof is to use Fried’s theory of suspension flow and dynamic blow-up of Mosher. If the time permits, we also discuss the non-existence of the analog of Fried’s continuous extension of the normalized entropy over the fibered face in the case of asymptotic translation lengths on the curve complex. This talk is based on joint work with Eiko Kin, Hyunshik Shin and Chenxi Wu. | |||||
16.30 - 17.30 | Zoom | Federico Vigolo | Asymptotic expander graphs (Slides) | ||
A sequence of expanders is a family of finite graphs that are sparse yet highly connected. Such families of graphs are fundamental object that found a wealth of applications throughout mathematics and computer science. This talk is centred around an "asymptotic" weakening of the notion of expansion. The original motivation for this asymptotic notion comes from the study of operator algebras associated with metric spaces. Further motivation comes from some recent works which established a connection between asymptotic expansion and strongly ergodic actions. I will give a non-technical introduction to this topic, highlighting the relations with usual expanders and group actions. | |||||
26/06/2020 | 14.00 - 15.00 | Zoom | Tianyi Zheng | Neretin groups admit no non-trivial invariant random subgroups (Slides) | |
We explain the proof that Neretin groups have no nontrivial ergodic invariant random subgroups (IRS). Equivalently, any non-trivial ergodic p.m.p. action of Neretin’s group is essentially free. This property can be thought of as simplicity in the sense of measurable dynamics; while Neretin groups were known to be abstractly simple by a result of Kapoudjian. The heart of the proof is a “double commutator” lemma for IRSs of elliptic subgroups. | |||||
16.00 - 17.00 | Zoom | Hiroki Matui | Various examples of topological full groups (Slides) | ||
I will begin with the definition of topological full groups and explain various examples of them. The topological full group arising from a minimal homeomorphism on a Cantor set gave the first example of finitely generated simple groups that are amenable and infinite. The topological full groups of one-sided shifts of finite type are viewed as generalization of the Higman-Thompson groups. Based on these two fundamental examples, I will discuss recent development of the study around topological full groups. | |||||
05/06/2020 | 15.00 - 16.00 | Zoom | Federico Berlai | From hyperbolicity to hierarchical hyperbolicity | |
Hierarchically hyperbolic groups (HHGs) and spaces are recently-introduced generalisations of (Gromov-) hyperbolic groups and spaces. Other examples of HHGs include mapping class groups, right-angled Artin/Coxeter groups, and many groups acting properly and cocompactly on CAT(0) cube complexes. After a substantial introduction and motivation, I will present a combination theorem for hierarchically hyperbolic groups. As a corollary, any graph product of finitely many HHGs is itself a HHG. Joint work with B. Robbio. | |||||
16.30 - 17.00 | Zoom | Mark Hagen | Hierarchical hyperbolicity from actions on simplicial complexes | ||
The notion of a "hierarchically hyperbolic space/group" grows out of geometric similarities between CAT(0) cubical groups and mapping class groups. Hierarchical hyperbolicity is a "coarse nonpositive curvature" property that is more restrictive than acylindrical hyperbolicity but general enough to include many of the usual suspects in geometric group theory. The class of hierarchically hyperbolic groups is also closed under various procedures for constructing new groups from old, and the theory can be used, for example, to bound the asymptotic dimension and to study quasi-isometric rigidity for various groups. One disadvantage of the theory is that the definition --- which is coarse-geometric and just an abstraction of properties of mapping class groups and cube complexes --- is complicated. We therefore present a comparatively simple sufficient condition for a group to be hierarchically hyperbolic, in terms of an action on a hyperbolic simplicial complex. I will discuss some applications of this criterion to mapping class groups and (non-right-angled) Artin groups. This is joint work with Jason Behrstock, Alexandre Martin, and Alessandro Sisto. | |||||
15/05/2020 | 15.00 - 16.00 | Zoom | Alex Bishop | Geodesic Growth in Virtually Abelian Groups | |
Bridson, Burillo, Elder and Šunić asked if there exists a group with intermediate geodesic growth and if there is a characterisation of groups with polynomial geodesic growth. Towards these questions, they showed that there is no nilpotent group with intermediate geodesic growth, and they provided a sufficient condition for a virtually abelian group to have polynomial geodesic growth. In this talk, we take the next step in this study and show that the geodesic growth for a finitely generated virtually abelian group is either polynomial or exponential; and that the generating function of this geodesic growth series is holonomic, and rational in the polynomial growth case. To obtain this result, we will make use of the combinatorial properties of the class of linearly constrained language as studied by Massazza. In addition, we show that the language of geodesics of a virtually abelian group is blind multicounter. | |||||
16.30 - 17.00 | Zoom | James East | Presentations for tensor categories | ||
Many well-known families of groups and semigroups have natural categorical analogues: e.g., full transformation categories, symmetric inverse categories, as well as categories of partitions, Brauer/Temperley-Lieb diagrams, braids and vines. This talk discusses presentations (by generators and relations) for such categories, utilising additional tensor/monoidal operations. The methods are quite general, and apply to a wide class of (strict) tensor categories with one-sided units. | |||||
01/05/2020 | 15.00 - 16.00 | Zoom | Yeeka Yau | Minimal automata for Coxeter groups | |
In their celebrated 1993 paper, Brink and Howlett proved that all finitely generated Coxeter groups are automatic. In particular, they constructed a finite state automaton recognising the language of reduced words in a Coxeter group. This automaton is not minimal in general, and recently Christophe Hohlweg, Philippe Nadeau and Nathan Williams stated a conjectural criteria for the minimality. In this talk we will explain these concepts, and outline the proof of the conjecture of Hohlweg, Nadeau, and Williams. We will also describe an alternative algorithm to minimise any finite state automaton recognising the language of reduced words in a Coxeter group, which utilises the associated root system of the group. This work is joint with James Parkinson. |
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16.30 - 17.30 | Zoom | Adam Piggott | The automorphism groups of the easiest infinite groups still present many mysteries (Slides) | ||
Free groups, and free products of finite groups, are the easiest non-abelian infinite groups to think about. Yet the automorphism groups of such groups still present significant mysteries. We discuss a program of research concerning automorphisms of easily understood infinite groups. | |||||
06/03/2020 | 10.00 - 11.00 | V 107 | Waltraud Lederle | Conjugacy and dynamics in the almost automorphism group of a tree | |
We define the almost automorphism group of a regular tree, also known as Neretin's group, and determine when two elements are conjugate. (joint work with Gil Goffer) | |||||
11.30 - 12.30 | V 107 | Mark Pengitore | Translation-like actions on nilpotent groups | ||
Whyte introduced translation-like actions of groups as a geometric generalization of subgroup containment. He then proved a geometric reformulation of the von Neumann conjecture by demonstrating that a finitely generated group is non amenable if and only it admits a translation-like action by a non-abelian free group. This provides motivation for the study of what groups can act translation-like on other groups. As a consequence of Gromov’s polynomial growth theorem, virtually nilpotent groups can act translation-like on other nilpotent groups. We demonstrate that if two nilpotent groups have the same growth, but non-isomorphic Carnot completions, then they can't act translation-like on each other. (joint work with David Cohen) | |||||
14.00 - 15.00 | V 107 | Jeroen Schillewaert | Fixed points for group actions on $2$-dimensional affine buildings | ||
We prove a local-to-global result for fixed points of groups acting on $2$-dimensional affine buildings (possibly non-discrete, and not of type $\tilde{G}_{2}$). In the discrete case, our theorem establishes two conjectures by Marquis. (joint work with Koen Struyve and Anne Thomas) | |||||
15.30 - 16.30 | V 107 | Francois Thilmany | Lattices of minimal covolume in $\mathrm{SL}_n$ | ||
A classical result of Siegel asserts that the (2,3,7)-triangle group attains the smallest covolume among lattices of $\mathrm{SL}_2(\mathbb{R})$. In general, given a semisimple Lie group $G$ over some local field $F$, one may ask which lattices in $G$ attain the smallest covolume. A complete answer to this question seems out of reach at the moment; nevertheless, many steps have been made in the last decades. Inspired by Siegel's result, Lubotzky determined that a lattice of minimal covolume in $\mathrm{SL}_2(F)$ with $F=\mathbb{F}_q((t))$ is given by the so-called characteristic $p$ modular group $\mathrm{SL}_2(\mathbb{F}_q[1/t])$. He noted that, in contrast with Siegel’s lattice, the quotient by $\mathrm{SL}_2(\mathbb{F}_q[1/t])$ was not compact, and asked what the typical situation should be: "for a semisimple Lie group over a local field, is a lattice of minimal covolume a cocompact or nonuniform lattice?". In the talk, we will review some of the known results, and then discuss the case of $\mathrm{SL}_n(\mathbb{R})$ for $n > 2$. It turns out that, up to automorphism, the unique lattice of minimal covolume in $\mathrm{SL}_n(\mathbb{R})$ ($n > 2$) is $\mathrm{SL}_n(\mathbb{Z})$. In particular, it is not uniform, giving a partial answer to Lubotzky’s question in this case. |
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01/11/2019 | 12.00 - 13.00 | V 109 | Anthony Dooley | Classification of non-singular systems and critical dimension | |
A non-singular measurable dynamical system is a measure space $X$ whose measure $\mu$ has the property that $\mu $ and $\mu \circ T$ are equivalent measures (in the sense that they have the same sets of measure zero). Here $T$ is a bimeasurable invertible transformation of $X$. The basic building blocks are the \emph{ergodic} measures. Von Neumann proposed a classification of non-singular ergodic dynamical systems, and this has been elaborated subsequently by Krieger, Connes and others. This work has deep connections with C*-algebras. I will describe some work of myself, collaborators and students which explore the classification of dynamical systems from the point of view of measure theory. In particular, we have recently been exploring the notion of critical dimension, a study of the rate of growth of sums of Radon-Nikodym derivatives $\Sigma_{k=1}^n \frac{d\mu \circ T^k}{d\mu}$. Recently, we have been replacing the single transformation $T$ with a group acting on the space $X$. | |||||
14.00 - 15.00 | V 109 | Colin Reid | Piecewise powers of a minimal homeomorphism of the Cantor set | ||
Let $X$ be the Cantor set and let $g$ be a minimal homeomorphism of $X$ (that is, every orbit is dense). Then the topological full group $\tau[g]$ of $g$ consists of all homeomorphisms $h$ of $X$ that act 'piecewise' as powers of $g$, in other words, $X$ can be partitioned into finitely many clopen pieces $X_1,...,X_n$ such that for each $i$, $h$ acts on $X_i$ as a constant power of $g$. Such groups have attracted considerable interest in dynamical systems and group theory, for instance they characterize the homeomorphism up to flip conjugacy (Giordano-Putnam-Skau) and they provided the first known examples of infinite finitely generated simple amenable groups (Juschenko--Monod). My talk is motivated by the following question: given $h\in\tau[g]$ for some minimal homeomorphism $g$, what can the closures of orbits of $h$ look like? Certainly $h\in\tau[g]$ is not minimal in general, but it turns out to be quite close to being minimal, in the following sense: there is a decomposition of $X$ into finitely many clopen invariant pieces, such that on each piece $h$ acts a homeomorphism that is either minimal or of finite order. Moreover, on each of the minimal parts of $h$, then either $h$ or $h^{-1}$ has a 'positive drift' with respect to the orbits of $g$; in fact, it can be written in a canonical way as a conjugate of a product of induced transformations (aka first return maps) of $g$. No background knowledge of topological full groups is required; I will introduce all the necessary concepts in the talk. | |||||
15.30 - 16.30 | V 109 | Michael Barnsley | Dynamics on Fractals | ||
I will outline a new theory of fractal tilings. The approach uses graph iterated function systems (IFS) and centers on underlying symbolic shift spaces. These provide a zero dimensional representation of the intricate relationship between shift dynamics on fractals and renormalization dynamics on spaces of tilings. The ideas I will describe unify, simplify, and substantially extend key concepts in foundational papers by Solomyak, Anderson and Putnam, and others. In effect, IFS theory on the one hand, and self-similar tiling theory on the other, are unified. The work presented is largely new and has not yet been submitted for publication. It is joint work with Andrew Vince (UFL) and Louisa Barnsley. The presentation will include links to detailed notes. The figures illustrate 2d fractal tilings. By way of recommended background reading I mention the following awardwinning paper: M. F. Barnsley, A. Vince, Self-similar polygonal tilings, Amer. Math. Monthly 124 (1017) 905-921. |
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06/09/2019 | 14.30 - 15.30 | V 109 | Sasha Fish | Patterns in sets of positive density in trees and buildings | |
We will discuss what are the patterns that necessarily occur in sets of positive density in homogeneous trees and certain affine buildings. Based on joint work with Michael Bjorklund (Chalmers) and James Parkinson (University of Sydney). | |||||
16.00 - 17.00 | V 109 | Michael Coons | Integer sequences, asymptotics and diffraction | ||
For some time now, I have been trying to understand the intricacy and complexity of integer sequences from a variety of different viewpoints and at least at some level trying to reconcile these viewpoints. However vague that sounds - and it certainly is vague to me - in this talk I hope to explain this sentiment a bit. While a variety of results will be considered, I will focus closely on two examples of wider interest, the Thue-Morse sequence and the set of $k$-free integers. | |||||
02/08/2019 | 12.00 - 13.00 | V 109 | Brian Alspach | Honeycomb Toroidal Graphs | |
The honeycomb toroidal graphs are a family of graphs I have been looking at now and then for thirty years. I shall discuss an ongoing project dealing with hamiltonicity as well as some of their properties which have recently interested the computer architecture community. | |||||
14.00 - 15.00 | V 109 | John Bamberg | Symmetric finite generalised polygons | ||
Finite generalised polygons are the rank 2 irreducible spherical buildings, and include projective planes and the generalised quadrangles, hexagons, and octagons. Since the early work of Ostrom and Wagner on the automorphism groups of finite projective planes, there has been great interest in what the automorphism groups of generalised polygons can be, and in particular, whether it is possible to classify generalised polygons with a prescribed symmetry condition. For example, the finite Moufang polygons are the 'classical' examples by a theorem of Fong and Seitz (1973-1974) (and the infinite examples were classified in the work of Tits and Weiss (2002)). In this talk, we give an overview of some recent results on the study of symmetric finite generalised polygons, and in particular, on the work of the speaker with Cai Heng Li and Eric Swartz. | |||||
15.30 - 16.30 | V 109 | Marston Conder | Edge-transitive graphs and maps | ||
In this talk I'll describe some recent discoveries about edge-transitive graphs and edge-transitive maps. These are objects that have received relatively little attention compared with their vertex-transitive and arc-transitive siblings. First I will explain a new approach (taken in joint work with Gabriel Verret) to finding all edge-transitive graphs of small order, using single and double actions of transitive permutation groups. This has resulted in the determination of all edge-transitive graphs of order up to 47 (the best possible just now, because the transitive groups of degree 48 are not known), and bipartite edge-transitive graphs of order up to 63. It also led us to the answer to a 1967 question by Folkman about the valency-to-order ratio for regular graphs that are edge- but not vertex-transitive. Then I'll describe some recent work on edge-transitive maps, helped along by workshops at Oaxaca and Banff in 2017. I'll explain how such maps fall into 14 natural classes (two of which are the classes of regular and chiral maps), and how graphs in each class may be constructed and analysed. This will include the answers to some 18-year-old questions by Širáň, Tucker and Watkins about the existence of particular kinds of such maps on orientable and non-orientable surfaces. |
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03/05/2019 | 14.00 - 15.00 | W 238 | 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. | |||||
15.30 - 16.30 | W 238 | Youming Qiao | Isomorphism testing problems: in light of Babai's graph isomorphism breakthrough | ||
In computer science, an isomorphism testing problem asks whether two objects are in the same orbit under a group action. The most famous problem of this type has been the graph isomorphism problem. In late 2015, L. Babai announced a quasipolynomial-time algorithm for the graph isomorphism problem, which is widely regarded as a breakthrough in theoretical computer science. This leads to a natural question, that is, which isomorphism testing problems should naturally draw our attention for further exploration? | |||||
12.00 - 13.00 | W 238 | 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. | |||||
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. |
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In this edition, three 2017 ARC Laureate Fellows in mathematics outline their projects. Descriptions of these projects are given as abstracts below. |
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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. |