the previous proposition.

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

Suppose $\phi$ is induced by a pseudo-Anosov. By Thurston’s hyperbolization theorem, $F*_\phi$ is the fundamental group of a complete finite-volume hyperbolic 3-manifold with cusps. In particular, Schwartz proved such groups are [*q.i.-rigid*]{}. As $F*_\phi$ and $F'*_\psi$ are quasi-isometric torsion-free groups, q.i.-rigidity of $F*_\phi$ implies they are commensurable. Thus $\psi$ is induced by a pseudo-Anosov by Proposition \[geomcomm\].

This proof underscores how difficult it is to prove the q.i.-invariance conjecture since there is no common model like $\mathbb H^3$ when studying irreducible and atoroidal endomorphisms.

Irreducible and Atoroidal Monodromies {#nongeom}
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The goal of this section is to prove that being irreducible and atoroidal is a commensurability invariant. We proved the following result in previous work [@JPM]; it essentially characterizes being an irreducible and atoroidal endomorphism.

\[invSbgrp\] Suppose $\phi : F \to F$ is an irreducible and atoroidal endomorphism. If nontrivial $K \le F$ is finitely generated and $i_x\phi^n(K) \le K$ for some $n \ge 1$ and inner automorphism $i_x$, 