Speakers

• Emmanuel Breuillard (University of Oxford) [Clay Lecturer]

Title: Free subgroups, expanders and random groups

Abstracts:

Lecture 1: This lecture will be devoted to the Tits alternative, its refinements and uniform versions, as well as some of its consequences regarding analysis and probability on groups.

Lecture 2: In the second lecture I will discuss the geometry and spectrum of Cayley graphs of finite simple groups of Lie type over a finite field, in particular the spectral gap/expander property and how it connects to the first lecture.

Lecture 3: In the third lecture I will turn attention to random groups in the few relator model and discuss recent joint with O. Becker and P. Varju in which we use the expander property of finite quotients to describe the character variety of a random group with values in a semi-simple Lie group.

• Martin Bridson (University of Oxford)

Title: Profinite rigidity, fibre products, and 3-manifolds

Lecture 1: An introduction to profinite rigidity

Abstract: To what extent is a finitely generated group determined by its set of finite quotients (equivalently, its profinite completion)? This compelling question has re-emerged with different emphases throughout the history of group theory, and in recent years it has been animated by a rich interplay with geometry and low-dimensional topology. In this lecture I will describe some highlights of the mathematics motivated by this question, distinguish between different forms of it, and describe some of the core challenges that are driving the field today. I shall also introduce a variety of invariants and ideas that can be used to analyse and distinguish between profinite completions of discrete groups. The lecture will be full of examples.

Lecture 2: Fibre products, finiteness properties, and Grothendieck pairs

Abstract: I shall explain why fibre products are a rich source of finitely presented groups and illustrate how they can be used to produce a diverse universe of pairs of finitely presented, residually finite groups H<G such that H is of infinite index but the inclusion of H into G induces an isomorphism of profinite completions – i.e. H<G is a Grothendieck pair. (The homology of groups plays a prominent role in this discussion.) I shall explain how variants of the theory of hyperbolic groups allow one to construct uncountably many pairs H<G with G fixed and finitely presented while the finiteness properties of H vary. I shall use this construction to introduce recent joint work with Reid and Spitler in which we construct the first examples of groups that are profinitely rigid in the universe of finitely presented groups, but not in the universe of finitely generated groups.

Lecture 3: Profinite rigidity and 3-manifolds

Abstract: I shall begin by recalling some of the special features that distinguish 3-dimensional manifolds from manifolds in other dimensions – above all, their geometry, arithmetic invariants, and the dominant role of the fundamental group. I shall then attempt to explain the state of the art concerning the extent to which we can recognise 3-manifolds from the finite images of their fundamental groups.

• Benson Farb (University of Chicago)

Title: Mapping class groups of K3 surfaces from a Thurstonian viewpoint

Abstract: In many ways the state of our understanding of homeomorphisms of 4-manifolds in 2022 is essentially that of our understanding of homeomorphisms of 2-manifolds in 1973, before Thurston changed everything. In these talks I will report on some first steps in a project (joint with Eduard Looijenga) whose ultimate goal is to change this. I will try to highlight some of the many open problems on this topic.

This mathematics involves a mix of ideas, from algebraic geometry to geometric topology to the theory of quadratic forms to reflection groups. The talks should be understandable to anyone who has taken first-year grad courses in algebraic and differential topology; some knowledge of basic algebraic geometry would be useful but is not essential.

• Nicolas Monod (EPFL)

Title: Three lectures around amenability

Lecture 1: An invitation to amenability

In 1929, John von Neumann analysed "miraculous duplication" paradoxes such as the Banach-Tarski paradox. He isolated the active ingredient involved in these psychedelic manifestations: a group-theoretical property that we now call "amenability". (Or to be quite precise: the negation thereof.) The first lecture will introduce a few faces of this fascinating concept blessed with multiple personalities. Examples, counter-examples and relations to geometry, analysis or combinatorics will be seen along the way.

Lecture 2: An indiscrete Bieberbach theorem

The meeting ground between non-positive curvature and amenability is shaped by flatness. This principle emerged in the 1970s and culminated in 1998 with a remarkable metric strengthening of the Bieberbach theorem: the flat Euclidean spaces are the only non-positively curved spaces admitting an amenable crystallographic group. If we allow continuous symmetries instead of only the discrete crystallographic ones, then the landscape becomes much more scenic; it includes trees, buildings and symmetric spaces. In joint work with Caprace, we have shown that the resulting geometries can still be classified in this indiscrete setting.

Lecture 3: Amenability for those who are not

The work of Furstenberg, Margulis and Zimmer has brilliantly illustrated that amenability is an invaluable tool to study non-amenable groups. The underlying concepts are those of "boundaries" and "amenable actions". We will indicate two recent examples where we used these objects to investigate very diverse groups: Gelfand pairs in one case, lamplighters in the other.

• Anne Thomas (University of Sydney)

Title: Large-scale geometry of right-angled Coxeter groups

Abstract: We introduce right-angled Coxeter groups and the spaces on which they act, and discuss several large-scale geometric concepts in this setting, including hyperbolicity and boundaries. We then explain Bowditch's JSJ tree, which records the key features of the boundaries of certain hyperbolic groups. We provide an explicit construction of Bowditch's JSJ tree for a class of right-angled Coxeter groups, then identify a subclass for which Bowditch's JSJ tree is a complete quasi-isometry invariant, hence the quasi-isometry classification of this subclass is decidable. This is joint work with Pallavi Dani.

• Amie Wilkinson (University of Chicago) [Clay Lecturer]

Title: Dynamical symmetry groups

Abstract: In classical mechanics, symmetry occurs for a reason: there is a conserved quantity such as angular momentum. This is Noether’s theorem, and it points to a broader theme in dynamics that symmetry is rare and meaningful. I will discuss, in the contexts of modern dynamics and geometry, how this theme recurs in beautiful ways: on the one hand, a typical object has the minimum amount of symmetry possible, and on the other hand, a little extra symmetry implies a lot of symmetry, a phenomenon known as rigidity.