[*induced by a pseudo-Anosov (on a punctured surface)*]{}, [*atoroidal*]{}, and [*irreducible*]{} (Section \[defs\]). We originally set out to prove that irreducibility was a group invariant of the automorphism’s mapping torus and, along the way, we proved more general statements for the first property and the composite property of being both irreducible and atoroidal. Our first result is that the first property is a geometric invariant:

[Theorem]{}[geomqi]{} Suppose $\phi: F \to F$ and $\psi:F' \to F'$ are free group injective endomorphisms such that the mapping tori $F *_\phi$ 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.

Thus starting with just a free group automorphism $\phi$ induced by a pseudo-Anosov and a quasi-isometry between $F \rtimes_\phi \mathbb Z$ and $F' *_\psi$, we find that $\psi$ is induced by a surface homeomorphism too. The proof is short but uses deep geometric theorems: Thurston’s hyperbolization [@Thu82] and Schwartz rigidity [@Sch95]. Since pseudo-Anosovs have dynamics that are very similar to those of irreducible and atoroidal automorphisms, it is likely that the latter property is a geometric invariant too.

Suppose $\phi : F \to F$ and $\psi : F' \to F'$ are 