tori if they have vanishing Euler characteristic.

\[hwlem\] Let $F$ be a free group of finite rank $m$ and $A \le F$ a free factor of rank $n$. Then $\chi(F*_{A}) = n - m$.

Choose a basis for $A$ and extend it to a basis for $F$. Then the natural finite presentation complex $X$ is aspherical, i.e., $X$ is a $K(\pi,1)$ space for $F*_{A}$, and $$\chi(F*_{A}) = \chi(X) = 1 - (1+m) + n = n-m. \qedhere$$

Finally, we give a characterization of irreducible endomorphisms with nontrivial periodic conjugacy classes. As the techniques used in the proof of the following theorem are not related to the rest of the paper, we postpone the proof to the appendix (Proposition \[eg\] and Theorem \[thmBH2\]).

\[thmBH\] Let $\phi:F \to F$ be an infinite-order endomorphism. Then $\phi$ is irreducible and has a periodic nontrivial conjugacy class if and only if it is induced by a pseudo-Anosov homeomorphism of a surface $\Sigma_g^b$ that acts transitively on the boundary components.

This theorem allows us to partition injective endomorphisms of free groups with interesting dynamics into two categories:

-   automorphisms induced by pseudo-Anosov maps.

-   irreducible and atoroidal endomorphisms.

Pseudo-Anosov Monodromies {#geom}
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The first result in 