I will present a new notion of non-positively curved groups: the collection of discrete countable groups acting (AU-)acylindrically on finite products of hyperbolic spaces. This work (joint with T. Fernos) is inspired by the classical theory of S-arithmetic lattices and that of acylindrically hyperbolic groups. I will start with the motivation for studying these groups in this talk, and cover some results we have proved that highlight this duality.

A median structure on a topological space is a continuous choice of a "median" of a triple of points satisfying some natural axioms. The standard example is provided by a CAT(0) cube complex, where the median is the unique point that lies on an ℓ¹ geodesic between any two of the three points. There are easy examples of median structures on ℝ² that don't arise from a cubing. However, we show that for many spaces, including ℝⁿ, every median structure is locally defined by a cubing. We study the converse, whether a collection of compatible local cubings defines a median structure, and this leads to the notion of a web structure, which is basically a collection of foliations. This work is ongoing and is joint with Ken Bromberg and Michah Sageev.

One can learn a lot about a compact manifold if one can show that it fibres over the circle — in essence, this allows us to view a static n-dimensional manifold as a manifold of dimension n−1 that evolves in time.

Being fibred (over the circle) is a relatively rare property. It is much more common to be virtually fibred, that is, to admit a finite cover that is fibred. For example, it was the content of a conjecture of William Thurston, now two theorems by Ian Agol and Dani Wise, that all finite-volume hyperbolic 3-manifolds are virtually fibred; in fact, this property is extremely common among irreducible 3-manifolds.

The situation is less clear in higher dimensions. On the obstruction side, we know that virtually fibred manifolds must have vanishing Euler characteristic. This immediately shows that compact hyperbolic manifolds in even dimensions will not be virtually fibred. A more involved obstruction comes from L²-homology: virtually fibred manifolds must be L²-acyclic.

The motivation behind the research I will present lies in trying to find situations in which the vanishing of L²-homology is not only necessary, but also sufficient for virtual fibring. It turns out that a lot more can be said if we replace aspherical manifolds by their homological cousins: Poincaré duality groups. Concretely, if G is an n-dimensional Poincaré-duality group over the rationals, and if G satisfies the RFRS property, then G is L²-acyclic if and only if there is a finite-index subgroup G₀ of G and an epimorphism from G₀ onto the integers such that its kernel is a Poincaré-duality group over the rationals of dimension n−1. (This last theorem is joint with Sam Fisher and Giovanni Italiano.)

The RFRS property was introduced in Agol's work on Thurston's conjecture. A countable group is RFRS if and only if it is residually {virtually abelian and poly-ℤ}. All compact special groups in the sense of Haglund-Wise satisfy this property, so there is a ready supply of RFRS groups, also among fundamental groups of hyperbolic manifolds in high dimensions.

An ongoing theme in mathematics, in general, and in group theory, in particular, is to study complicated objects by approximating them by simpler ones. In group theory, this led to notions like residually finite, sofic, hyper-linear, etc. We describe some of the major problems in this area, including both progress and failures.

Profinite rigidity refers to the property of a finitely generated, residually finite group being completely determined by its profinite completion. In other words, a finitely generated residually finite group G is profinitely rigid if, whenever a finitely generated residually finite group H has the same profinite completion as G, then H is isomorphic to G. This talk will discuss progress on the question of profinite rigidity amongst groups arising in low-dimensional geometry and topology. This will include a discussion of recent history, ongoing work as well as posing questions to hopefully stimulate further research in the area.

Given a group G and a collection of subgroups of G, we consider the quasi-isometries of G that coarsely preserve the collection of cosets of these subgroups. Together these quasi-isometries form the relative quasi-isometry group. We are interested in cases where this relative quasi-isometry group is “discrete”, or even isomorphic to the original group G. After discussing some relevant background and motivation, I will present two new results about this, one for free groups and one for Bourdon lattices.