Schedule and abstracts

Mark Hagen - Cube complexes and groups

The mini-course is on nonpositively-curved cube complexes, particularly special cube complexes and their uses.  We will start with right-angled Artin and Coxeter groups and discuss why it is advantageous to try to (virtually) embed one's favourite group into a RAAG.  We will then discuss special cube complexes, how the definition enables one to construct an embedding of the fundamental group in a RAAG, and the combinatorial methods for constructing finite covers (generalising what one can do in graphs) that are available in the setting of special cube complexes.  We will also discuss specific examples, like 3-manifolds and mapping tori of certain free group automorphisms.

Kathryn Mann - Groups, 3-manifolds, and Anosov-like actions

This mini-course centers on the broad theme of the intricate relationship between the algebraic structure of the group, the dynamics of this group acting on a space, and geometric structures related to such actions. A paradigm example of this is the  "convergence group theorem" of Tukia, Gabai, and Casson-Jungries, which says that convergence groups (a purely dynamical condition) acting on the circle are always Fuchsian groups, i.e isomorphic to the fundamental group of a hyperbolic surface or orbifold; and the action on the circle is the action on the boundary of the group (equivalently on the boundary of the universal cover of the orbifold).  

In a similar spirit, Thurston defined an "extended convergence group",  an analogous dynamical condition for groups acting on the line, and showed using ideas of Fenley that these are always fundamental groups of 3-manifolds and the action on the line comes from a dynamical system on the 3-manifold called an Anosov flow.   However, not all 3-manifolds with Anosov flows give rise to extended convergence groups; instead giving rise to what are called Anosov-like actions on bifoliated planes. This minicourse will present a broad perspective on the theme of reconstructing groups and geometry from actions on spaces, aiming towards recent work on describing groups with Anosov-like actions on planes.  

Thomas Koberda - Using logic to study homeomorphism groups

I will describe some recent results on the first order rigidity of homeomorphism groups of compact manifolds, and their applications to dynamics of group actions on manifolds. I will also describe how to find "syntactic" invariants of manifolds, and how these can be used to give a conjectural model-theoretic characterization of the genus of a surface.

Bruno Martelli - Hyperbolic manifolds in higher dimension

We will briefly recall the different techniques available to build hyperbolic manifolds in dimension 2 and 3, and then move to dimension n>=4 where the tools available are essentially two: Coxeter polytopes and arithmetic groups. The main focus will be to understand their topology.

Chloe Perin - Model theory and geometric group theory

The model theory of an algebraic structure concerns itself with first-order formulas, which can be thought of as a generalization of equations. At the beginning of the millenium, using tools from geometric group theory, Zlil Sela proved that free groups of different ranks are indistinguishable from the point of view of model theory, thus solving the long standing Tarski problem. Since then, geometric techniques have been used to solve a number of questions coming from model theory, and have provided a fascinating new lens through which to look at our favorite groups. I will give an overview of this interaction and of the surprising connections that appear in this context.  I will assume no background in model theory.

Macarena Arenas - (Strong) asphericity for (cubical) group presentations

A cell complex X with fundamental group G is a classifying space for G if it is aspherical. Such spaces are of interest in geometric group theory because, in a sense, their geometry is the closest to that of the groups they are associated with. Every group has a classifying space, and all classifying spaces for a given group are homotopy equivalent, but finding "nice" or "useful" classifying spaces is often difficult.

An ideal situation is when a group G is given by a presentation P with some nice property, and the complex associated to this presentation is aspherical. A generalisation of this arises in the setting of cubulated groups, and is related to studying certain quotients, called cubical presentations, which encode geometric information in a similar way to the classical group presentations.

In this talk we will explore some classes of groups that admit aspherical (classical or cubical) presentations, and explain how a strong form of asphericity, called the Cohen-Lyndon property, naturally arises in connection to these groups. Time permitting, we will discuss some applications.

Inhyeok Choi - Genericity of pseudo-Anosov mapping classes

The celebrated Nielsen-Thurston classification concerns mapping classes of a finite-type hyperbolic surface. Among three categories, pseudo-Anosov mapping classes exhibit the most interesting dynamics and serve as central objects in 2- and 3-dimensional topology. Nonetheless, it is not obvious that pseudo-Anosov mapping class even exists at all. After the existence was known by Thurston's construction, it has been conjectured that pseudo-Anosov mapping classes are generic in mapping class groups. In this talk, I will talk about some progresses on this genericity problem. 

Francesco Fournier-Facio - Hyperbolic structures on Thompson's group F

When a group does not exhibit too much negative curvature, it is sometimes possible to completely describe its cobounded actions on hyperbolic spaces, which are encoded in a poset. This is the case for Thompson's groups T (circle) and V (Cantor set). I will report on ongoing work with Sahana Balasubramanya and Matt Zaremsky, where we attempt to do this for Thompson's group F (interval). The corresponding poset turns out to be huge in every possible sense of the word, but it still exhibits some regularities that allow for a detailed description.