then $[F : (i_x \phi^n)^{-k}(K)] < \infty$ for some $k \ge 0$.

The key idea in this section is to use this proposition to characterize in terms of the mapping torus exactly when a monodromy is irreducible and atoroidal. To this end, we need the following property to deal with nonsurjective monodromies.

An injective endomorphism $\phi:F \to F$ is [**minimal**]{} if the image $\phi(F)$ is not contained in a proper free factor of $F$. Automorphisms and irreducible endomorphisms are clearly minimal.

Minimality is preserved by taking powers and composing with (inner) automorphisms (See [@JPM Proposition 5.4]). We now have enough to state and prove the main result:

\[grpinv\] Suppose $\phi: F \to F$ is a minimal injective endomorphism and let $G = F*_\phi$. Then $\phi$ is irreducible and atoroidal if and only if $G$ has no finitely generated noncyclic subgroups with infinite index and vanishing Euler characteristic.

If $\phi$ is not atoroidal, then $G$ has a $\mathbb Z^2$ subgroup that necessarily has infinite index. If $\phi$ is not irreducible, then there exists a proper free factor $A \le F$, $x \in F$, and $n \ge 1$ such that $\phi^n(A) \le xAx^{-1}$. Then, using normal forms, $A *_{i_x \phi^n} \cong \langle 