mini-courses

Macarena Arenas

A crash course on isoperimetry

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.

Eduardo Oregón-Reyes

Cubulated groups and virtual specialness

CAT(0) cube complexes were introduced by Gromov merely as examples of metric spaces of non-positive curvature, but now they play a prominent role in geometric group theory. One reason for this is that many interesting groups are known to act nicely on these spaces, including free and surface groups, small cancellation groups, 1-relator groups with torsion, and many 3 manifold groups. Another reason is that some of these groups are, in addition, virtually special, notion defined by Haglund and Wise that implies being (up to finite index) the subgroup of some right-angled Artin group.

In the first lecture, we will define CAT(0) cube complexes, explore some of their combinatorial structure, and discuss some examples of cubulated groups. For the second lecture, we will introduce the class of virtually special groups, review some of their properties, and mention some criteria for virtual specialness. We will end the mini-course with a discussion of the main techniques for studying cubulated hyperbolic groups, focussing on some theorems of Wise and Agol. If time permits, I will mention a few things about the relatively hyperbolic case.

Joshua Frisch

The Poisson boundary and bounded harmonic functions on groups

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.

Konstantinos Tsouvalas

Anosov representations of hyperbolic groups

Anosov representations form a rich and stable class of discrete subgroups of reductive Lie groups. They were introduced by Labourie in his study of the Hitchin component for fundamental groups of closed negatively curved Riemannian manifolds and further generalized by Guichard-Wienhard for more general hyperbolic groups.

In the first lecture, we are going to provide the dynamical definition of an Anosov representation and some equivalent definitions. In the second lecture and part of the third lecture, we are going to discuss the main properties of Anosov representations and their limit maps and provide some classes of examples. In the third lecture, we will also discuss a construction of linear hyperbolic groups (joint work with Nicolas Tholozan) which fail to admit discrete faithful representations into simple Lie groups of real rank 1.

Sahana Balasubramanya

Acylindrically and relatively hyperbolic groups

The class of acylindrically hyperbolic groups has been of immense interest in recent times. It is an extremely large class of groups, containing many interesting examples. Yet a significant part of the theory of hyperbolic and relatively hyperbolic groups can be generalized in this context. The goal of this mini course is to provide an introduction to this class of groups, and focus on some important techniques.

In the first lecture, we will define acylindrical actions and talk about the motivation to study them. We will then define acylindrically hyperbolic groups and discuss some examples and properties of this class of groups. The second lecture will focus on the notion of hyperbolically embedded subgroups and discuss relative hyperbolicity in this context. In the last lecture, we will discuss the concept of group theoretic Dehn filling and some of its applications. Time permitting, I will also talk a bit about my own research.

Rose Morris-Wright

Artin groups: algebraic and geometric techniques

Artin groups are a broad class of groups whose presentations all follow a particular pattern. They are generalizations of braid groups and are closely related to Coxeter groups. Artin groups provide examples of groups with many interesting properties but there is very little that is known about ALL Artin groups.

In the first lecture, we’ll define Artin groups, talk about different types of Artin groups, and give a summary of known results and open questions. In the second lecture, we’ll focus on algebraic techniques for studying Artin groups, including the Garside structure, parabolic subgroups, and, if time permits, the Artin monoid. In the third lecture, we’ll discuss geometric techniques for studying Artin groups, including the Deligne complex and newer complexes such as the Clique-cube complex and the systolic Artin complex.

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