Topology Seminar

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.

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.

I will discuss a uniform waist inequality in codimension 2 for the family of finite covers of a Riemannian manifold whose fundamental group has Kazhdan‘s property T. I will describe a general strategy to prove waist inequalities based on a higher property T for Banach spaces. The general strategy can be implemented in codimension 2 but is conjectural in higher codimension. We speculate about the situation for lattices in semisimple Lie groups. Based on joint work with Uri Bader