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) 

The theory of JSJ decomposition plays a key role in the classification of hyperbolic groups, in analogy with the case of 3-manifolds. While this theory can be extended to larger families of groups, the JSJ decomposition displays significant flexibility in general, making a complete understanding of its behaviour more challenging. In this talk, Dario Ascari explores this flexibility, with an emphasis on the case of generalized Baumslag-Solitar groups.