only if the monodromy is atoroidal and 2) a word-hyperbolic free-by-cyclic group has a menger curve [*Gromov boundary*]{} if and only if the monodromy fixes no free splitting of the fiber; Macura [@Mac02] proved that two polynomially-growing free group automorphisms that have quasi-isometric mapping tori must have polynomial growth of the same degree and, conjecturally, these mapping tori cannot be quasi-isometric to a mapping torus of an exponentially growing free group automorphism.

[**Outline.**]{} We give the standard definitions and statements of results that are most relevant to the rest of the article in Section \[defs\]. Section \[geom\] contains the proof of Theorem \[geomqi\] while Section \[nongeom\] contains that of Theorem \[nongeomcomm\]. Finally, we briefly discuss the q.i.-invariance conjecture in Section \[qi\]. In Appendix \[app\], we prove a folk theorem and its converse: a free group endomorphism is irreducible with a nontrivial periodic conjugacy class if and only if it is induced by a pseudo-Anosov on a punctured surface with one orbit of punctures.

**Acknowledgments:** I want to thank Ilya Kapovich for patiently checking my initial proof (and its numerous iterations) and Chris Leininger for sharing with me his work-in-progress in collaboration with Spencer Dowdall and I. Kapovich as it relates 