Program

Schedule

Abstracts

Minicourses

Volodymyr Nekrashevych: Groups of dynamical origin

Abstract: We will talk about groups naturally associated with dynamical systems: iterated monodromy groups, full groups of homeomorphisms, full groups of etale groupoids, etc... The techniques involved in the study of properties of these groups are usually different from the "classical" group theory and use topological dynamics and geometry of orbital graphs. We will see how these techniques help to study groups, in particular, such properties as torsion, growth, and amenability.

Piotr Przytytcki: The isomorphism problem for Coxeter groups

Abstract: When are two Coxeter groups isomorphic? We discuss a conjectural answer proposed by Bernhard Muehlherr. We outline the solution in “twist-rigid” case that is joint work with Pierre-Emmanuel Caprace.

Stefan Wenger: Dehn functions and large scale geometry

Abstract: The Dehn function of a space measures how difficult it is to fill a curve of given length by a disc. It is an important invariant used in analysis, geometry and especially in geometric group theory where it is a quasi-isometry invariant of the group and connected to the difficulty of solving the word problem. The growth of the Dehn function is closely related to the large scale geometry of the underlying space. For example, Gromov hyperbolic spaces are exactly those with a linear Dehn function. In contrast, the large scale geometry of spaces with quadratic Dehn function is not well understood yet and only few properties shared by all such spaces are known. Spaces with (at most) quadratic Dehn function include spaces of non-positive curvature on the one hand and many spaces with a mix of negative and positive curvature on the other hand, for example the higher Heisenberg groups and other nilpotent groups.

The goal of the minicourse is to give an introduction to the Dehn function. with particular emphasis on the study of relationships between the growth of Dehn functions and large scale properties of the underlying space or group. Asymptotic cones will play a crucial role in our considerations as will be an analytic variant of the Dehn function - the Sobolev Dehn function. This is particularly useful in the study of spaces with quadratic Dehn function as it allows to establish a link between the growth of the Dehn function and the fine geometry of asymptotic cones. We will consider various classes of spaces including Gromov hyperbolic spaces, nilpotent groups, and spaces with quadratic Dehn function.

If time permits I will furthermore briefly discuss how the Sobolev Dehn function can be used to give a new proof of the important Bonk-Kleiner theorem on quasisymmetric parametrizations of metric 2-spheres. This theorem is interesting in connection with Cannon's conjecture in geometric group theory which asserts that if the boundary at infinity of a Gromov hyperbolic group is homeomorphic to the 2-sphere then it is even quasisymmetric to the 2-sphere.

Research talks

Carolyn Abbott: Acylindrical actions on hyperbolic spaces

Abstract: The class of acylindrically hyperbolic groups consists of groups that admit a particular nice type of non-elementary action on a hyperbolic space, called an acylindrical action. This class contains many interesting groups such as non-exceptional mapping class groups, Out(Fn) for n > 1, and right-angled Artin and Coxeter groups, among many others. Such groups admit uncountably many different acylindrical actions on hyperbolic spaces, and one can ask how these actions relate to each other. In this talk, I will describe how to put a partial order on the set of acylindrical actions of a given group on hyperbolic spaces, which roughly corresponds to how much information about the group different actions provide. This partial order organizes these actions into a poset. I will give some structural properties of this poset and, in particular, discuss for which (classes of) groups the poset contains a largest element.

Federico Berlai: A refined combination theorem for hierarchically hyperbolic groups

Abstract: Hierarchically hyperbolic groups and spaces are newly introduced classes that generalise Gromov-hyperbolicity, introduced in 2015 by J. Behrstock, M. Hagen, and A. Sisto. They include mapping class groups, right-angled Artin groups, and groups acting properly cocompactly on CAT(0) cube complexes.

In this talk, after an introduction and motivation, I will describe structural results for morphisms between hierarchically hyperbolic spaces, and a (new) combination theorem for hierarchically hyperbolic groups.

Joint work with Bruno Robbio.

Jingyin Huang: The asymptotic geometry of 2-dimenional Artin groups

Abstract: We show that 2-dimensional Artin groups are non-positively curved in the sense that they act geometrically on spaces whose minimal disks are CAT(0). As an application, we study the structure of quasi-Euclidean tilings over these Artin groups and deduce some quasi- isometric rigidity results for them. This is joint work with D. Osajda.

Nir Lazarovich: Detecting sphere boundaries of hyperbolic groups

Abstract: We show that the boundary of a one-ended simply connected at infinity hyperbolic group with enough codimension-1 surface subgroups is homeomorphic to a sphere. By works of Markovic and Kahn-Markovic our result gives a new characterization of groups which are virtually fundamental groups of hyperbolic 3-manifolds. I will discuss this joint work with Benjamin Beeker and further work in progress.

Arie Levit: Local rigidity of uniform lattices

Recall that a uniform lattice in a topological group is a discrete cocompact subgroup. I will talk about the fact that small deformations of a finitely generated uniform lattice must be isomorphic uniform lattices. Interestingly, the proof uses some geometric group theory techniques. The case of arbitrary topological groups is due to Gelander-L, generalizing Weil's classical theorem for Lie groups. I will also describe situations where the only possible deformations of a uniform lattice in the isometry group of a CAT(0) space are inner.

Alexandre Martin: On the automorphism group of cyclic products of groups

Graph products of groups are a construction that interpolates between direct products and free products, and contain well-known examples such as right-angled Coxeter groups and right-angled Artin groups. In this talk, I will present a form of rigidity for the automorphism group of `cyclic products of groups', that is, the graph products over cycles on at least 5 vertices. I will recall a construction due to Davis that allows us to understand graph products through their action on CAT(0) cube complexes, and explain how this action extends to the whole automorphism group in the case of cyclic products of at least 5 groups. Such an action can be used to completely compute their automorphism group and to show their acylindrical hyperbolicity. This is joint work with Anthony Genevois.

Jenny Wilson: Stability in the homology of configuration spaces

Abstract: This talk will illustrate some patterns in the homology of the configuration space F_k(M), the space of ordered k-tuples of distinct points in a manifold M. For a fixed manifold M, as k increases, we might expect the topology of these configuration spaces to become increasingly complicated. Church and others showed, however, that when M is connected and open, there is a representation-theoretic sense in which the homology groups of these spaces stabilize. In this talk I will explain these stability patterns, and describe higher-order stability phenomena – relationships between unstable homology classes in different degrees – established in recent work joint with Jeremy Miller. This project was inspired by work-in-progress of Galatius–Kupers–Randal-Williams.