Seminars in Combinatorial Group Theory

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Robert Gray: Subgroups of inverse monoids via the geometry of their Cayley graphs

4th May, 2023

In the 1960s Higman was able to characterize the finitely generated subgroups of finitely presented groups, that is, groups defined using a finite set of generators and finite set of defining relations. His result, which is called the Higman Embedding Theorem, is a key result in combinatorial group theory which makes precise the connection between group presentations and logic. In this talk I will present a result of a similar flavour, proved in recent joint work with Mark Kambites (Manchester), in which we characterise the groups of units of inverse monoids defined by presentation where all the defining relators are of the form w=1. I will explain what an inverse monoid is, the motivation for studying this class of inverse monoids, and also outline some of the geometric ideas that we developed in order to prove our results.

Pavel Zalesski: Combinatorial theory of pro-p groups

15th December, 2022

Free products with amalgamation and HNN-extensions are two main constructions of combinatorial group theory. I shall discuss these two constructions in the category of pro-p groups, presenting results on splittings of pro-p groups as an amalgamated free pro-p product or a pro-p HNN-extension and relating them with pro-p version of Bass-Serre's theory of groups acting on trees. I shall also compare the pro-p results with similar results for abstract groups.

Murray Elder: On groups presented by inverse-closed finite convergent length-reducing rewriting systems

18th February, 2022

In the 1980s Madlener and Otto asked for an algebraic characterisation of groups presented by finite, convergent, length-reducing rewriting systems, conjecturing that they are exactly the plain groups (free product of finitely many finite groups and infinite cyclic groups).

I will describe some recent results with Adam Piggott (ANU) on new geometric, algebraic and algorithmic properties of groups presented by (inverse-closed) finite, convergent, length-reducing rewriting systems.

Alexei Myasnikov: On the Andrews-Curtis conjecture

8th October, 2021

I am going to talk about the group-theoretic aspects of the Andrews-Curtis conjecture, some recent results, and some old. From my viewpoint the Andrews-Curtis conjecture is not just a hard stand-alone question, coming from topology, but a host of very interesting problems in group theory.

Ilya Kapovich: Non-linear words and free groups

29th April, 2021

An important theme in the study of combinatorics of words involves looking for models of nonlinear words, that is words that are not indexed by segments of integers. We discuss one such model arising from the theory of Stallings subgroup graphs. This model naturally leads to the notion of subset currents on free groups (and on other word-hyperbolic groups) which are measure-theoretic analogs of conjugacy classes of finitely generated subgroups. Many new features manifest themselves in this context, including connections with the Hanna Neumann Conjecture and Whitehead's algorithm for subgroups.

Tim Susse: Geometric Properties of Random Right-angled Coxeter Groups

15th April, 2021

Given a finite simplicial graph, we can generate a right-angled Coxeter group (RACG): each vertex corresponds to a generator and two generators commute if and only if the corresponding vertices are adjacent. This assignment is unique up to isomorphism, which allows us to characterize geometric and algebraic properties of the RACG using combinatorial properties of its generating graph. This also allows us to use a model of random graphs to study random RACGs.

In this talk we will focus on the divergence of a RACG. Using the Erdös-Renyi random graph model, we show that at a wide range of densities a random RACG asymptotically almost surely has quadratic divergence. We will also give a sharp threshold, below which a random RACG has (almost surely) at least cubic divergence, and above which its divergence is (almost surely) at most quadratic. This is joint work with Jason Behrstock, Victor Falgas-Ravry and Mark Hagen.

Murray Elder: Rewriting systems and geodetic graphs

4th September, 2020

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.