Handlebody bundles and polytopes (Q2070095)

The authors construct an infinite family of non-diffeomorphic three-manifolds with first Betti number \(\geq 2\) and which fiber over the circle such that all the monodromies of the fibered faces of the corresponding Thurston polytope extend from the closed surface on which they are defined to a handlebody. The proof of this result relies on a connection between handlebody bundles and free-by-cyclic groups. The result establishes a connection between mapping classes of surfaces and outer automorphisms of free groups, namely, to every such automorphism one can associate (infinitely many) mapping classes which inherit properties from the free group automorphism.