Schedule

(Last updated May 6th)

Minicourses

Hyungryul Baik: Introduction to laminar groups

A group acting on the circle with invariant laminations is called a laminar group. Laminar groups naturally arise in low-dimensional topology. All surface groups and 3-manifold groups with interesting structures such as taut foliations, Anosov or pseudo-Anosov flows, and veering triangulations are examples of laminar groups.

Laminar group actions of 3-manifold groups were first constructed by W. Thurston under the presence of co-orientable taut foliations and his work largely influenced a lot of authors including D. Calegari, N. Dunfield, S. Fenley, S. Frankel, K. Mann, and myself.

In the mini-course, I will first talk about the construction of invariant laminations from these structures and then discuss reconstruction results, namely constructing 2 or 3-dimensional manifolds with interesting structures from laminar group actions.

Koji Fujiwara: Weakly acylindrical group actions on hyperbolic spaces

I will discuss weakly acylindrical actions of finitely generated groups on hyperbolic spaces.

I will explain that discrete actions of finitely generated groups on H^n are weakly acylindrical in a certain uniform sense.

As an application, I will discuss the Tits alternative and the growth. 

If time permits, I will present some uniform estimates for the drift of random walks on these groups.

Autumn Kent: Surface bundles, convex cocompactness, and hyperbolicity.

I will give a rather broad overview and introduction to the theory of convex cocompactness in mapping class groups in the sense of Farb and Mosher and its relation to the geometry of groups and surface bundles.

Jing Tao: Title: Introduction to mapping class groups, big and small

This mini-course introduces mapping class groups of both finite-type (“small”) and infinite-type (“big”) surfaces. We will discuss basic examples and key geometric features, with particular emphasis on the Nielsen–Thurston classification in the classical finite-type setting, and on the ways the infinite-type world departs from this picture. 

Research Talks

Monika Kudlinska: Conjugacy separability in free-by-cyclic groups

A group is free-by-cyclic if it admits a homomorphism onto an infinite cyclic group with free kernel. Free-by-cyclic groups exhibit a range of interesting geometric and algebraic phenomena which are often governed by the dynamics of the conjugation action on the kernel. The aim of this talk is to discuss the new result that all finitely generated free-by-cyclic groups are conjugacy separable; that is, for any pair of non-conjugate elements, there exists a finite quotient in which the images remain non-conjugate. The proof involves several staple geometric group theory techniques, including analysis of actions on trees and Dehn filling. I will provide a gentle introduction to these tools, which form an essential part of any budding geometric group theorist's toolkit. This is based on joint work with Francois Dahmani, Sam Hughes and Nicholas Touikan.

Marco Linton: Hanna Neumann bounds for cubulated locally quasi-convex hyperbolic groups.

The Hanna Neumann conjecture predicted that the reduced rank of the intersection of two finitely generated subgroups H and K of a free group F is bounded above by the product of the reduced rank of H with the reduced rank of K. After more than 50 years had passed, Mineyev and Friedmann successfully (and independently) turned the Hanna Neumann conjecture into a theorem. In this talk I will explain some recent work with Sam Hughes in which we manage to generalise this to the setting of torsion-free cubulated locally quasi-convex hyperbolic groups. The notion of reduced rank will need to be replaced with the notion of reduced Euler characteristic, but otherwise the statement will turn out to be identical. I will conclude by explaining some interesting consequences for the class of free-by-cyclic groups.

Yulan Qing: Large Scale geometry of big mapping class groups

We construct a curve graph consisting of non-peripheral, simple closed curves for surfaces of infinite type. We show that it is connected and often hyperbolic. We use the non-peripheral curve graph to show that a large set of big mapping class groups have at most quadratic divergence. A section application is that we prove a large set of big mapping class groups have infinite coarse rank. This talk is based on joint work with Kasra Rafi and Assaf Bar-Natan. If time permits, we will discuss recent work joint with Alex Squires; we show that geodesics in non-peripheral curve graphs also satisfy the bounded geodesic image theorem.

Xiaolei Wu: Embedding groups into acyclic groups 

We will start the talk with the definition of homology of groups. Then we discuss various constructions of acyclic groups in the literature and how one can embed a group of type F_n into an acyclic group of type $F_n$. The embedding we have uses the labeled Thompson group which goes back to Thompson's Splinter group in the 1980s. We explain how one can show that the labeled Thompson groups are always acyclic. This also allows us to build the first acyclic groups that are of type $F_n$ but not $F_{n+1}$ for any $n \geq 2$. If time permits, I will also discuss related results in the simple setting using the twisted Brin--Thompson groups. This is based on a joint work with Martin Palmer.

Abdul Zalloum: Sageev's construction of CAT(0) cube complexes and beyond

Sageev's revolutionary work shows that CAT(0) cube complexes can be fully described via their hyperplanes and the way these interact with each other. Starting with an arbitrary set and a collection of bipartitions, called walls, Sageev’s construction provides a canonical way to build a CAT(0) cube complex from such data. I will discuss joint work with Petyt showing that Sageev’s construction extends to spaces and groups that are not necessarily cubulated, and often yields results strikingly similar to those for CAT(0) cube complexes. Several applications will be discussed.