Abstracts

Mini-courses

Radhika Gupta: Introduction to free-by-cyclic groups

A free-by-cyclic group is an extension of a free group by an infinite cyclic group. This is a rich class of groups and has close connections to the study of automorphisms of free groups. The aim of this mini-course will be to give an introduction to free-by-cyclic groups and survey some important results in this area. In the first lecture, we will start with some basic properties of free-by-cyclic groups. In the second lecture, we will see when these groups are (relatively) hyperbolic and admit actions on CAT(0) cube complexes. In the third lecture, I will give an introduction to the 'fibered face theory' in analogy to Thurston's theory for fibered 3-manifolds. Finally, we will conclude with some open questions about free-by-cyclic groups. 

Thomas Haettel: Group actions on Helly graphs and injective metric spaces

We will discuss Helly graphs and injective metric spaces: basic definitions, elementary properties, simple examples. We will focus on nonpositive curvature features, such as local properties, bicombings and classification of isometries. We will present numerous groups acting geometrically by isometries on such spaces, such as hyperbolic groups, braid groups, mapping class groups and lattices in Lie groups.

Clara Löh: L2-invariants of groups

There is a rich interaction between group-theoretic properties and homological invariants of groups and spaces. Taking twisted coefficients in L2-functions on a group (or related constructs such as the group von Neumann algebra), leads to homological invariants of groups that, e.g., contain information  on homology growth of groups along finite index subgroups and on dynamical properties of groups.

This mini-course will give a brief introduction to L2-Betti numbers of groups, their computation, and applications.

Brita Nucinkis: Cohomological finiteness properties for totally disconnected locally compact groups

The aim of this course is to introduce cohomological finiteness conditions both from an algebraic and a geometric viewpoint. I will begin by introducing the main concepts, such as cohomological dimension and cohomological finiteness lengths for discrete groups, and give an overview of some interesting examples. I will also introduce Brown’s criterion, which is a powerful tool to determine the cohomological finiteness length. Many of these concepts can be extended to study finiteness conditions for totally disconnected locally compact groups. I will give an overview of current results and open questions.

Plenary talks

Amandine Escalier: Measure and orbit equivalence of graph products

Graph products were introduced by Green in 1990 and generalise both right-angled Artin groups and right-angled Coxeter groups. Like the latter two, graph products act nicely on CAT(0) cube complexes, making them particularly appealing objects for geometric group theorists.

Although CAT(0) cube complexes and geometric group theory are absolutely fantastic, here, we will study graph products from the equally fantastic point of view of measured dynamics. All necessary prerequisites will be recalled, the words “measured groupoids” might be pronounced (but only to give a name to colourful pictures), and parallels with geometric group theory will be made.

This (the results presented, not the abstract) is a joint work with Camille Horbez.

Elia Fioravanti: Automorphisms of virtually special groups

Mapping class groups, Out(F_n) and GL_n(Z) are three important groups in geometric group theory that arise as outer automorphism groups of other groups (respectively, of surface, free and free abelian groups). In general, it can be rather complicated to describe the structure of the group Out(G) for more general groups G, particularly if only certain properties of G are known. I will outline some of the work of Rips and Sela in the 90s, which completely describes Out(G) for any Gromov-hyperbolic group G, based on a canonical JSJ decomposition of G. Then I will talk about a new canonical JSJ decomposition for the compact special groups of Haglund and Wise, and how this can be used to gain information on Out(G) for any virtually special group G.

Swathi Krishna: Cannon–Thurston maps

Given a hyperbolic group and a hyperbolic subgroup, if the inclusion extends continuously to a map between their Gromov boundaries, the boundary map is called the Cannon–Thurston map. In this talk, we will first briefly look at the cases where the Cannon–Thurston map exists. Then we focus on this particular result: let 1 → N → G→ Q → 1 be an exact sequence of non-elementary hyperbolic groups. By the work of Mahan Mj, we know that N → G admits a Cannon–Thurston map. Let Q′ be a quasi-isometrically embedded subgroup of Q. We will see that the pre-image H of Q′ in G is hyperbolic and the inclusion H → G admits the Cannon–Thurston map.

This is based on a joint work with Pranab Sardar.

Eduardo Reyes: Green metrics and metric structures on hyperbolic groups

Teichmüller space is a classical construction that, for a given closed hyperbolic surface, parameterizes the geometric actions of its fundamental group on the hyperbolic plane. I will talk about a generalization of this space, where for an arbitrary hyperbolic group we consider a metric space that parameterizes its geometric actions on Gromov hyperbolic spaces. Even in the surface group case, this space turns out to be much larger than Teichmüller space, and we can find points induced by negatively curved Riemannian metrics, Anosov representations, random walks, geometric cubulations, etc. In particular, I will discuss how Green metrics (those encoding admissible random walks on the group) are dense in this space. This is joint work with Stephen Cantrell and Dídac Martínez-Granado.

Lorenzo Ruffoni: The fundamental groups of 3-pseudomanifolds

The study of 3-manifolds has been a major source of inspiration for geometric group theory, providing the initial motivation behind many definitions and theorems. More generally, 3-pseudomanifolds are polyhedral complexes that look like 3-manifolds almost everywhere, but can have some singular vertices whose links are surfaces of positive genus. These spaces naturally arise in the study of certain Coxeter groups, but in general the class of fundamental groups of 3-pseudomanifolds remains quite mysterious. In this talk we will discuss similarities and differences between hyperbolic 3-manifolds and hyperbolic 3-pseudomanifolds, focusing on problems such as computing the L2-homology and finding nice isometric actions.

Research statements