The lecturers are:
Recordings of these 6 lectures as well as those for other networks are now available from an LMS administered site.
We also have slides and lecture notes for Marialaura's first lecture, slides and lecture notes for Marialaura's second lecture,
slides and annotated slides for Alan's first lecture, slides and annotated slides for Alan's second lecture,
slides and annotated slides for Alex' first lecture, slides and annotated slides for Alex' second lecture.
1. Introduction to growth in groups, by Alex Evetts (Newcastle)
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.
2. Free groups via graphs, by Alan Logan (Heriot-Watt)
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.
3. Groups of automorphisms of rooted trees, by Marialaura Noce (Goettingen)
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 ele- mentary amenable group. Furthermore, this group provides a counterexample to the General Burnside Problem.
In this lecture 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.