holds as well, i.e., a hyperbolic 3-manifold that fibers over a circle will have a pseudo-Anosov [*monodromy*]{}, the property of a fibered manifold having a pseudo-Anosov monodromy is in fact a geometric invariant: if $f:S\to S$ and $f':S' \to S'$ are homeomorphisms whose mapping tori have [*quasi-isometric*]{} (q.i.) fundamental groups, then $f$ is pseudo-Anosov if and only if $f'$ is pseudo-Anosov.

There are three types of invariants that we study in geometric group theory: group invariants, which contain virtual/commensurability invariants, which contain geometric/q.i.-invariants; the geometric invariants are the most important and difficult to prove. In this paper, we exhibit geometric and commensurability invariants for free-by-cyclic groups inspired by Thurston’s hyperbolization theorem and our arguments will be general enough to also apply to ascending HNN extensions of free groups.

There is a rough correspondence between the study of the outer automorphism group of a free group $\operatorname{Out}(F)$ and the study of the mapping class group of a hyperbolic surface $\operatorname{MCG}(S)$. Under this correspondence, surface groups are paired with free groups, surfaces with graphs, and 3-manifolds that fiber over a circle with free-by-cyclic groups. However, this correspondence is not perfect; pseudo-Anosov mapping classes have three possible analogous properties for free group automorphisms: 