of the isomorphisms $\epsilon\left.\phi^i\right|_{K}: K \to K'$ and $\left.\psi^j \epsilon\right|_{K}:K \to K'$ are the same for some $i, j \ge 1$; a priori, we restrict ourselves to $i, j$ such that $\phi^i(K) = K$ and $\psi^j(K') = K'$, i.e., the maps $\epsilon\left.\phi^i\right|_{K}$ and $\left.\psi^j \epsilon\right|_{K}$ make sense. Masai-Mineyama prove that in this case, $\phi$ is irreducible and atoroidal if and only if $\psi$ is too. In other words, being irreducible and atoroidal is a fibered commensurability invariant. However, compared to commensurability, this equivalence is very restrictive since a typical isomorphism of finite index subgroups of free-by-cylic groups will not preserve the fibers one starts with.

Feighn-Handel’s theorem that mapping tori of free group endomorphisms are coherent is the main tool that allows us to avoid the obstacles in the two approaches discussed above. We shall explicitly use the [*preferred presentation*]{} that their algorithm produces for any finitely generated subgroup of a mapping torus (Theorem \[rewrite\]).

We conclude the introduction by noting that very little is known about geometric invariants of free-by-cyclic groups in relation to their monodromies. Here are the geometric invariants we know of: Brinkmann [@Bri00; @Bri02] showed 1) a free-by-cyclic group is word-hyperbolic (a geometric invariant) if and 