combine this theorem with Corollary \[irredtorgrp\] to get:

\[irredgrp\]Suppose $\phi: F \to F$ and $\psi:F' \to F'$ are minimal injective endomorphisms such that $F *_\phi \cong F' *_\psi$. Then $\phi$ is irreducible if and only if $\psi$ is irreducible.

Q.I.-Invariance Conjecture {#qi}
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Very little is known about the geometry of mapping tori whose monodromies are irreducible and atoroidal. Brinkmann proved that atoroidal automorphisms of free groups have word-hyperbolic mapping tori [@Bri00] and gave an explicit description for when such mapping tori have non-trivial splittings over cyclic subgroups [@Bri02]: an atoroidal automorphism of $F$ has a mapping torus that splits over $\mathbb Z$ if and only if the (outer) automorphism fixes a [*free splitting*]{} of $F$; in particular, mapping tori of irreducible and atoroidal automorphisms never split over $\mathbb Z$.

By work of Kapovich-Kleiner [@KK00], word-hyperbolic free-by-cyclic groups that do not split over $\mathbb Z$ have Gromov boundary homeomorphic to a menger curve. Unfortunately, the topology of the boundary is not sufficient to detect irreducible and atoroidal monodromies as there are reducible and atoroidal automorphisms that do not fix a free splitting.

The Gromov boundary of a word-hyperbolic group can be given a [*visual metric*]{} that is unique up to 