peripheral ones. Thus $\psi$ is induced by an infinite-order irreducible homeomorphism. By Nielsen-Thurston classification, $\psi$ is induced by a pseudo-Anosov.

Since the number of orbits of boundary components for a pseudo-Anosov is the number of boundary components in the mapping torus, this proposition combines with Theorem \[thmBH\] to give:

\[irredtorgrp\] Suppose $\phi: F \to F$ and $\psi:F' \to F'$ are free group injective endomorphisms such that $F *_\phi \cong F' *_\psi$. Then $\phi$ is irreducible and has a periodic nontrivial conjugacy class if and only if $\psi$ is irreducible and has a periodic nontrivial conjugacy class.

Surprisingly, the analogous statement for commensurable groups is much harder to prove. The difficulty lies in showing that if the restriction of an iterate $\psi^n$ to a finite index subgroup is induced by a pseudo-Anosov, then so is $\psi$. We could use the theory of train tracks to adapt the argument in Appendix \[app\] and get a comparatively elementary proof; we opt to use Thurston’s hyperbolization [@Thu82] and Mostow’s rigidity [@Mar74] to keep the exposition short.

\[geomcomm\] Suppose $\phi: F \to F$ and $\psi:F' \to F'$ are free group injective endomorphisms such that $F *_\phi$ and $F' *_\psi$ are commensurable. Then $\phi$ is induced 