= \langle t^i F t^{-i}, t \rangle$.

The following lemma shows that the property of having a finitely generated noncyclic subgroups with infinite index and vanishing Euler characteristic is a commensurability invariant. This implies the invariance of irreducibility for atoroidal endomorphisms.

Let $G$ be a finitely generated torsion-free group and $G' \le G$ be a finite index subgroup. Then $G$ has an infinite index finitely generated noncyclic subgroup with vanishing Euler characteristic if and only if $G'$ has such a subgroup.

The reverse direction is obvious. Suppose $H \le G$ is an infinite index finitely generated noncyclic subgroup with vanishing Euler characteristic. Then $H \cap G'$ has finite index in $H$. In particular, it is finitely generated with infinite index in $G$ and hence $G'$. Furthermore, $[ H: H \cap G'] < \infty$ implies $H \cap G'$ has vanishing Euler characteristic too and it is noncyclic since the only virtually cyclic torsion-free group is $\mathbb Z$, yet $H$ is not cyclic.

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

Finally, we 