Fibring, foliations and group theory

The phenomena of 3-manifolds fibring over S^1 has strong links with group theory. A particular instance of this is Stallings’s fibring theorem, which roughly says that a compact 3-manifold fibres over S^1 if and only if its fundamental group admits a nontrivial homomorphism to Z with finitely generated kernel. A manifold fibring over S^1 is in some sense generalised by having a (codimension 1) foliation, with the latter forming a far broader class of objects. As such, one cannot hope in general to see a foliation in the fundamental group of your manifold, and especially not in as nice a form as a group homomorphism! In this talk we will give a gentle introduction to the objects mentioned above, before introducing a particularly nice class of foliations introduced by Thurston which do in fact appear in the fundamental group in the form of a quasimorphism with strong geometric properties. Time permitting, I will mention some ongoing work with Paula Heim on the study of these quasimorphisms from the perspective of group theory and coarse geometry.