this section is a straightforward application of Stallings’ fibration theorem and Nielsen-Thurston classification; it is the first half of the proof that irreducibility of the monodromy is a group invariant of the mapping torus. The subsequent generalizations are given to motivate the q.i.-invariance conjecture.

\[geomgrp\] Suppose $\phi: F \to F$ and $\psi:F' \to F'$ are free group injective endomorphisms such that $F *_\phi \cong F' *_\psi$. Then $\phi$ is induced by a pseudo-Anosov if and only if $\psi$ is induced by a pseudo-Anosov.

Without loss of generality, suppose $\phi$ is induced by a pseudo-Anosov on a compact surface with boundary. Therefore, $G = F \rtimes_\phi \mathbb Z$ is the fundamental group of a compact $3$-manifold that fibers over a circle. Bieri-Neumann-Strebel [@BNS] showed that the BNS-invariant of such a compact $3$-manifold group $G$ is symmetric, which implies that $\psi$ is an automorphism and $G \cong F' \rtimes_\psi \mathbb Z$. By Stallings’ fibration theorem [@St61], $\psi$ is induced by a homeomorphism of a compact surface with boundary. Any invariant essential multicurve of the $\psi$-inducing homeomorphism would determine a non-peripheral $\mathbb Z^2$ subgroup of $G$. But since $\phi$ was induced by a pseudo-Anosov, the only $\mathbb Z^2$ subgroups of $G$ are the 