mini-course descriptions

Radhika Gupta

Automorphisms from Outer space

Free groups are among the simplest groups occurring in nature. In order to understand free groups, we study their group of symmetries: the group of (outer) automorphisms of free groups Out(F_n). The study of Out(F_n) acquired a strong geometric flavor by the influence of Gromov and Thurston. Specifically, the introduction of Outer space, the equivalent of Teichmüller space, by Culler-Vogtmann and development of train track machinery by Bestvina-Feighn-Handel caused a spur of activity in the field. Later, the introduction of various hyperbolic complexes on which Out(F_n) acts also opened a lot of doors for exploration. In this minicourse, I will introduce Outer space, train track maps, folding paths in Outer space, and some hyperbolic complexes.

Robert Kropholler

A coherent introduction to incoherence

A group is coherent if every finitely generated subgroup is finitely presented. Examples of coherent groups include 3-manifold groups, free-by-cyclic groups, and polycyclic groups. More often, a group is incoherent (i.e., contains a finitely generated subgroup that is not finitely presented). Examples of incoherent groups include mapping class groups (g>1), Out(F_n) (n>2), Aut(F_n) (n>1), free-by-free groups, most RAAGs, ...

I will begin by discussing examples of incoherent groups and related finiteness properties. These will be the examples of Bieri and Stallings. These are subgroups of a product of free groups and thus provide an easy way to show incoherence in many situations. In the second talk, I will discuss how fibering can be used to show incoherence, allowing one to leverage recent theorems of Kielak to show incoherence in various settings including random groups. Finally, I will discuss how to prove a group is coherent. There are few known examples of coherent groups and I will try and provide details or ideas of how each is proved. I will end by discussing some open problems on the subject.

Marissa Kawehi Loving

An intro to mapping class groups

The mapping class group of a connected, oriented, finite-type surface S, denoted Mod(S), is the group of homotopy classes of orientation-preserving homeomorphisms of S. In this mini-course, we will cover some of the basics of Mod(S) such as building a generating set, producing actions of Mod(S) on various metric spaces (the curve complex, Teichmuller space, etc.), as well as building some (coarse) models for Mod(S).

In particular, we aim to build enough language and tools to give a brief overview of Masur-Minsky's hierarchy machinery by the end of the second lecture (this will hopefully give a good jumping off point for Davide's lectures on HHG's). Time allowing, in the final lecture, we will explore "big" mapping class groups of infinite-type surfaces, which is an area of increasing interest and is full of open questions and directions.

Rylee Lyman

Bass–Serre theory: groups acting on trees

Graphs can be cut up into forests along vertices. Surfaces can be cut up into disks along simple closed curves. Haken manifolds can be cut up into 3-balls along 2-sided, embedded incompressible surfaces. Bass–Serre theory provides a common framework for reasoning about cutting up a group along subgroups. As such it is perhaps the main inductive tool in geometric group theory. In this mini-course, we will meet the main features of the theory: graphs of groups, their fundamental groups, their universal covering trees, and their reformulation (due to Scott and Wall) as graphs of spaces. We will also study covering spaces and morphisms of graphs of groups, less well-known but no less useful. At the course we will meet a handful of applications of Bass–Serre theory. Possible examples include JSJ decompositions of hyperbolic groups, the "Nielsen realization problem" for finite groups of outer automorphisms of free groups, and the Bestvina–Feighn combination theorem.

Davide Spriano

Introduction to hierarchically hyperbolic groups

Hierarchically hyperbolic groups form a large class of groups that include, for instance, mapping class groups, many (conjecturally all) cubical groups, and many 3-manifold groups. Apart from containing examples of interest, there are powerful techniques that can be used to study hierarchically hyperbolic groups. These techniques allow one to prove theorems such as characterizations of acylindrical hyperbolicity and of hyperbolically embedded subgroups, results on uniform exponential growth, control over quasi-flats, dynamics on boundaries, and so on.

The goal of this mini-course is to provide a self-contained introduction to this theory. There will be roughly two parts. In the first part, we will motivate the idea of hierarchically hyperbolic groups and introduce the axiomatic definition. We will focus on several examples to motivate the various axioms. In the second part, we will focus on how to use the axioms to prove results about hierarchically hyperbolic groups. This will be done by studying results about growth of hierarchically hyperbolic groups using the structure of hierarchical hyperbolicity.

Federico Vigolo

Expander graphs

This mini-course is an introduction to the study of expander graphs. Informally, these are families of finite graphs that are sparse yet highly connected. Rather than giving a broad overview of the subject, I will focus on a few specific topics. Namely, error correcting codes and coarse embeddings. The former is a neat application of expander graphs in computer science; the latter is a topic of special interest for geometric group theorists which motivated the construction of Gromov's "monster groups". Coarse embeddings also give a good excuse to introduce the spectral characterization of expander graphs. In turn, this allows us to construct some explicit examples by means of Cayley graphs of quotients of a group with Kazhdan's Property (T).

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