I will relate two notorious open questions in low-dimensional topology.  The first asks whether every hyperbolic group is residually finite. The second, the congruence subgroup property, relates the finite-index subgroups of mapping class groups to the topology of the underlying surface. I will explain why, if every hyperbolic group is residually finite, then mapping class groups enjoy the congruence subgroup property. Time permitting, I may give some further applications to the question of whether hyperbolic 3-manifolds are determined by the finite quotients of their fundamental groups.

In the 1990s, Goodwillie developed a theory of calculus for homotopical functors. His idea was to approximate a functor by a tower of ‘polynomial functors’, similar to how one approximates a function by its Taylor series. The role of linear polynomials is played by functors that behave like homology theories, in the sense that there is a Mayer-Vietoris sequence computing their homotopy groups. As such, the Goodwillie tower interpolates between stable and unstable homotopy theory. The theory has application to the computation of the homotopy groups of spheres, higher algebra, and algebraic K-theory. In my talk, I will give an introduction to this topic. In particular, I will explain that Goodwillie's calculus reveals a deep connection between the homotopy theory of spaces and Lie algebras and how this is related to a chain rule for the derivatives of functors.
 

I will talk about the problem of recognising when a manifold admits a finite cover that fibres over the circle, with emphasis on the case of hyperbolic manifolds in odd dimensions. I will survey the state-of-art, and discuss the role that group theory plays in the problem. Finally, I will discuss a recent result that sheds light on the analogous group-theoretic problem, that is, virtual algebraic fibring of Poincaré-duality groups. The final theorem is joint with Sam Fisher and Giovanni Italiano.

Thompson’s groups, introduced by Thompson in 1965, have had a lot of attention in the last fifty years. Being finitely presented, a natural question is to compute their Dehn function. All three groups are conjectured to have quadratic Dehn function; this conjecture was confirmed for Thompson’s group 𝐹 by Guba in 2006. During Matteo Migliorini's talk, we show how to deduce from Guba’s result that Thompson’s group 𝑇 has a quadratic Dehn function as well.

I'll discuss how to show 3D Poincaré duality and residual finiteness are together incompatible with property (T).

We give examples to show that theorems of S Fisher and D Kielak 
relating the vanishing of L^2 cohomology to fibering over the 
circle for RFRS groups cannot be extended to some larger classes
of groups, and we introduce an L^2-torsion invariant that may 
prove useful. 

(Joint with Sam Hughes and Wolfgang Lueck)