Free-by-cyclic groups

In the 1980s William Thurston proposed a programme whose aim was to classify closed 3-manifolds. The two main steps where, first, to show that closed 3-manifolds can be glued from geometric pieces (Geometrisation Conjecture), and second, to show that the hyperbolic pieces, up to passing to a finite cover, arise only in one way, as mapping tori of homeomorphisms of surfaces of finite type (Virtually Fibred Conjecture).
Thurston's programme is now largely finished. The Geometrisation Conjecture was proven in 2003 by Grigoriy Perelman, including a positive solution of the Poincare conjecture. The Virtually Fibred Conjecture was solved in 2012 by Ian Agol. But the tools used to resolve the conjectures live on, and in this research project they will be applied to the class of free-by-cyclic groups.
Free-by-cyclic groups are (algebraic) mapping tori of free group automorphisms. The study of such automorphisms forms a backbone of geometric (and once of combinatorial) group theory; it is an extremely rich and exciting theory, with many open problems, multiple connections to other parts of mathematics, and a lot of current activity.
Classically, free-by-cyclic groups were understood primarily through the lens of automorphisms of free groups. This is very much analogous to studying a fibred 3-manifold through the lens of a fixed way in which the manifold fibres over the circle. What we would like to do in this project is to take a more holistic approach: to treat a free-by-cyclic group as a single object capable of many different descriptions as the mapping torus of a free group automorphisms. Then, as with fibred 3-manifolds, we can ask: what do the various automorphisms of free groups which produce the same free-by-cyclic group have in common?
This question is widely open. We know that if one of the automorphisms is atoroidal, that is, does not preserve a non-trivial conjugacy class, then the free-by-cyclic group is hyperbolic, which forces all the other automorphisms yielding the same free-by-cyclic group to also be atoroidal. Much more interestingly, it was shown in 2018 by Jean-Pierre Mutanguha that the same holds when atoroidal is replaced by being fully irreducible. Both of these results have an analogue on the 3-manifolds side - but there is only one analogue there! Being atoroidal and being fully irreducible both correspond, in the setting of surface bundles, to the surface homeomorphism being pseudo-Anosov. This exemplifies our approach: one can draw inspiration from what is known about 3-manifolds to form conjectures and attack-strategies in the vastly richer and more nuanced realm of free-by-cyclic groups. This is precisely what we intend to do.

This project falls within the EPSRC's "Geometry & Topology" research area, within the "Mathematical Sciences" theme; it has strong ties to the "Algebra" area as well. The aim of the project is twofold: to educate a new specialist in techniques vital in sustaining the core research capability of the UK; and to develop a high-impact toolkit combining ideas from geometry, algebra, and dynamics, with transformative potential for the theory of free-by-cyclic groups and the many areas of mathematics on which it impinges.