free group automorphisms such that $F \rtimes_\phi \mathbb Z$ and $F' \rtimes_\psi \mathbb Z$ are quasi-isometric. Then $\phi$ is irreducible and atoroidal if and only if $\psi$ is irreducible and atoroidal.

Our main result is that being irreducible and atoroidal is a commensurability invariant, which lends credence to the conjecture; again, the argument works for endomorphisms.

[Theorem]{}[nongeomcomm]{} Suppose $\phi: F \to F$ and $\psi:F' \to F'$ are free group injective endomorphisms such that $F *_\phi$ and $F' *_\psi$ are commensurable and neither one of the endomorphisms has an image contained in a proper free factor of their domain. Then $\phi$ is irreducible and atoroidal if and only if $\psi$ is irreducible and atoroidal.

The hypothesis on the images is necessary: Let $\phi:F_2\to F_2$ be the endomorphism on a free group of rank $2$ given by $\phi(a) = ab$ and $\phi(b) = ba$. Then $\phi$ is irreducible and atoroidal [@JPM Example 1.2]. Now let $F_2$ be a proper free factor of the free group $F_3$ generated by $\{ a,b,c \}$. Extend $\phi$ to $\psi:F_3 \to F_3$ by setting $\psi(c) \in F_2$; then $F_3*_{\psi} \cong F_2*_\phi$, but $\psi$ is reducible.

The proof of Theorem \[nongeomcomm\] follows immediately from an algebraic characterization of 