Fibering of 3-manifolds and free-by-cyclic groups

A 3-manifold fibers over the circle if it can be identified with the mapping torus of a surface homeomorphism. If the surface is compact with non-empty boundary then the corresponding 3-manifold group is free-by-cyclic, and the action of the cyclic group on the free group is induced by the surface homeomorphism. Although most free-by-cyclic groups do not arise as fundamental groups of 3-manifolds which fiber over the circle, there is a strong analogy between the two families.

In this talk I will discuss how dynamical properties of the monodromy affect the geometry/algebra of the corresponding mapping torus. We will see how the same 3-manifold or group can admit multiple fiberings and what properties of the monodromy are known to be preserved under different fiberings.