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Is the mapping torus of an automorphism of a free group virtually an amalgamated product?

Let $F$ be a nonabelian finitely generated free group, let $\tau \in \mathrm{Aut}(F)$ be an element of infinite order, and set $G = F \rtimes \mathbb{Z}$, where the action of $\mathbb{Z}$ on $F$ is ...
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