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-stabilizer on the corresponding neighbourhood. I will explain some of its importance and discuss some attempts to generalize it to the case of directed graphs.
The study of groups with "hyperbolic-like directions" has been a central theme in geometric group theory. Two notions are usually used to quantify what is meant by "hyperbolic-like directions", the notion of a contracting geodesic and that of a Morse geodesic. Since the property that every geodesic ray in metric space is contracting or Morse characterizes hyperbolic spaces, being a contracting/Morse geodesic is considered a hyperbolic-like property. Generalizing work of Cannon, I will discuss a joint result with Eike proving that for any finitely generated group, the language of contracting geodesics with a fixed parameter is a regular language. This immediately implies that contracting geodesics can't exist in torsion groups.
The Morse notion is a weaker notion than that of the contracting notion, in fact, building on work of Osin, Ol’shanskii, and Sapir, Fink gave an example of a torsion group which contains an infinite Morse geodesic. This seems to contradict the claim that Morse geodesics are "hyperbolic-like" directions. As an attempt to rectify this, Russell, Spriano, and Tran introduced a class of spaces where Morse (quasi)-geodesics satisfy some local-to-global property and they showed that many interesting examples live in such a class. In these spaces, Morse (quasi)-geodesics are expected to behave more reasonably like "hyperbolic directions", therefore, such spaces/groups can be regarded as good hosts of Morse (quasi)-geodesics.
I will discuss some continuation of their work where we show that in such spaces Morse geodesics form a regular language, give a characterization of stable subgroups in terms of regular languages. Time permitting, I will discuss few other applications of these automatic structures to the growth of stable subgroups and the dynamics of the action of such groups on their Morse boundaries. This work is joint with Cordes, Russell and Spriano.
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 characterized planar graphs as those that do not admit the complete graph on five 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.
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.
An LMS online lecture course in groups, semigroups and algebras.
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 non-discrete, 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.
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.
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 G 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 ε0 + 1. All except the maximum element of this family (which is F itself) are elementary amenable groups. In this way, for each α less than ε0, we obtain a finitely generated elementary amenable subgroup of F whose EA-class is α + 2. The talk will is pitched for an algebraically inclined audience, but little background knowledge will be assumed.
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.
We show that for almost all primitive integral cohomology classes in the fibred cone of a closed fibred hyperbolic 3-manifold, the monodromy normally generates the mapping class group of the fibre. 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 analogue of Fried’s continuous extension of the normalized entropy over the fibered face in the case of asymptotic translation lengths on the curve complex.
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