The Poisson boundary of a group has two interpretations. Firstly it measures the asymptotic uncertainty of a random walk on a group. Secondly it classifies the possible range of bounded harmonic functions on the group. In this minicourse we will introduce some of the theory of the Poisson boundary and its relationship with group properties such as polynomial growth, the infinite conjugacy class property, and amenability. In particular we will focus on the following question. For which measured groups are all bounded harmonic functions trivial? This will lead us into a, perhaps surprising, incredibly interconnected web of ideas including convex analysis, dynamical systems, information theory, and probability theory. No prior knowledge of random walks on groups or any of the aforementioned fields will be assumed.

The word problem for a finitely generated group G is the algorithmic problem of deciding whether a word in the generators represents the trivial element of G. When G is finitely presented, one can interpret this problem topologically by constructing a finite 2-complex X whose 1-cells and 2-cells correspond to the generators and relations. In this way, a word w in the generators which represents the trivial element of G will correspond to a loop in the 1-skeleton of the universal cover of X.

Geometry comes into the picture via the study of isoperimetric functions: An isoperimetric function is a function associated to a finite group presentation which bounds the area of a relation in that group in terms of the length of that relation (or, equivalently, the area of a nullhomotopic loop in the complex described above in terms of the length of that loop). A Dehn function is an optimal isoperimetric function. Dehn functions can be understood as quantifying the complexity of the word problem.

This mini-course will survey what is known about Dehn functions for various classes of groups.

In the first lecture, we will discuss the precise connection between the solvability of the word problem for a group and the growth of its Dehn function. We will compute the Dehn functions of various classes of groups (introducing, on the way, hyperbolic groups and CAT(0) groups) and give examples of groups with very large Dehn functions.

In the second lecture, we will dive more deeply into the structure of Dehn functions, exploring the "isoperimetric spectrum", i.e., the set {d | n^d is the growth type of a Dehn function}, and computing some more examples.

We will wrap things up by discussing some generalisations and variations of Dehn functions, and how they connect to other algorithmic problems in geometric group theory. Time permitting, I will also tell you a bit about my own work in this direction.

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.

An LMS online lecture course in profinite methods in geometric group theory.

An LMS online lecture course in Kazhdan's property (T).