$F*_\phi$ that detects exactly when $\phi: F \to F$ is irreducible and atoroidal.

[Theorem]{}[grpinv]{} Suppose $\phi: F \to F$ is a free group injective endomorphism whose image is not contained in a proper free factor of $F$ 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.

These results imply that irreducibility is a group invariant, our original motivation:

[Corollary]{}[irredgrp]{} Suppose $\phi: F \to F$ and $\psi:F' \to F'$ are free group injective endomorphisms such that $F *_\phi \cong F' *_\psi$ and neither one of the endomorphisms has an image contained in a proper free factor. Then $\phi$ is irreducible if and only if $\psi$ is irreducible.

That irreducibility is a group invariant was an open problem [@DKL17c Question 1.4]. In a series of papers [@DKL15; @DKL17b; @DKL17c], Dowdall-Kapovich-Leininger studied the dynamics of ([*word-hyperbolic*]{}) free-by-cyclic groups and the main result of the third paper answered this problem under an extra condition that we now discuss:

Fix a free-by-cyclic group $G$. The [*BNS-invariant*]{} $\mathcal C(G)$ is an open cone (with convex components) in $H^1(G; \mathbb R) \cong \operatorname{Hom}(G, \mathbb R)$. By rational 