Tag - Geometric group theory

George Domat: Free products from spinning and rotating families

19th November, 2020

A natural goal of geometric group theory is to understand the algebraic properties of a group via geometry. The far-reaching work of Dahmani-Guirardel-Osin and recent work of Clay-Mangahas-Margalit provide geometric approaches to the study of the normal closure of a subgroup in a large ambient group. In particular, their work gives conditions under which the normal closure is a free product. I will talk about recent work that aims to unify their results and gives a significantly shorter proof of the theorem of DGO. This is joint work with M. Bestvina, R. Dickmann, S. Kwak, P. Patel, and E. Stark.

Alex Evetts: Introduction to growth in groups

19th November, 2020

An LMS online lecture course in growth in groups.

For a finitely generated group, the number of elements that can be spelled with words of length n, for any integer n>0, is called the growth function. This can be interpreted as a measure of the size of the group and is a powerful quasi-isometry invariant which has links to many areas of geometric group theory.

In the first lecture I will present the fundamental properties of the growth function and explore some key examples illustrating what kinds of functions can arise. I will also discuss Gromov's important theorem on groups of polynomial growth.

In the second lecture I will discuss the formal power series associated to the growth function, which is known as the growth series. I will explain some ways in which the behaviour of the growth series can provide insight into the asymptotics, and demonstrate this with examples.

Doron Puder: Random permutations sampled by free words

16th November, 2020

Fix a word w in a free group on r generators. A w-random permutation in the symmetric group SN is obtained by sampling r independent uniformly random permutations σ1, . . .,σr ∈ SN and evaluating w(σ1, . . .,σr). Such w-random permutations have surprisingly rich structure with relation to deep results in geometric group theory. I'll survey some of this structure, state some conjectures, and explain how it is related to evaluating the spectral gap of random Schreier graphs of SN.

Wenhao Wang: Dehn Functions of Finitely Presented Metabelian Groups

12th November, 2020

The Dehn function was introduced by computer scientists Madlener and Otto to describe the complexity of the word problem of a group, and also by Gromov as a geometric invariant of finitely presented groups. In this talk, I will show that the upper bound of the Dehn function of finitely presented metabelian group G is 2n2k, where k is the torsion-free rank of the abelianization Gab, answering the question that if the Dehn functions of metabelian groups are uniformly bounded. I will also talk about the relative Dehn function of finitely generated metabelian group and its relation to the Dehn function.

Henry Bradford: Quantitative LEF and topological full groups

9th November, 2020

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.

Alan Logan: Free groups via graphs

5th November, 2020

An LMS online lecture course in free groups and graph theory.

Free groups may be viewed as the fundamental groups of graphs. This observation allows for a very intuitive view of free groups and their subgroups. These lectures combine topological ideas, due to Stallings in the 1980s, with more combinatorial and computational ones to prove many of the fundamental results in free groups. These results include the Nielsen-Schreier Theorem (subgroups of free groups are free), Howson's Theorem (finitely generated subgroups have finitely generated intersection), and the decidability of the subgroup membership problem.

Marialaura Noce: Groups of automorphisms of rooted trees

22nd October, 2020

An LMS online lecture course in groups acting on trees.

Groups of automorphisms of rooted trees have been studied for years as an important source of groups with interesting properties. For instance, the Grigorchuk group (that is a group acting on the binary tree) is the first example of a finitely generated group with intermediate growth (this answered an open question posed by Milnor) and the first example of an amenable but not elementary amenable group. Furthermore, this group provides a counterexample to the General Burnside Problem.

In these lectures we will first introduce the basic theory of groups of automorphisms of rooted trees and their subgroups. Then we will give examples and main properties of such groups, including the aforementioned Grigorchuk group, and the GGS groups.