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 given by $\tau$.
Must there be a finite index subgroup $U$ of $G$, 
such that $U \cong A*_CB$ where $C$ is a proper subgroup of both $A$ and $B$, and its index in either $A$ or $B$ is at least $3$?
What happens if $\tau$ is inner?
 A: This holds for atoroidal automorphisms by a result of Hagen and Wise. 
If $\tau$ is atoroidal, then $G$ is word-hyperbolic by a result of Bestvina and Handel. Since $G$ acts properly and cocompactly on a CAT(0) cube complex (by the above result), there is a finite-index subgroup for which the quotient of the action is a compact special cube complex. In particular, it has embedded walls (which are 2-sided). A bit of fiddling around gives the desired splitting. 
One of the walls induces a splitting of the group along a quasi-convex subgroup, but it might be an HNN extension, not an amalgamated product. We may as well assume that all of the walls are non-separating. Then one may pass to a double cover dual to a wall in which the preimage of the wall is now separating. To get an amalgam, connect the walls by an arc on one side. Then the fundamental group splits as an amalgamated product over this subgroup $C$ which is the free product of the fundamental group of the two walls ($A$ is the fundamental group of the half containing the arc, and $B$ is the fundamental group of the other side free product with $\mathbb{Z}$ corresponding to the arc). One can arrange that the amalgamated product satisfies your condition on $C$ by first passing to a sufficiently large cyclic cover dual to a wall. 
A: If you drop "virtually" from your question, then this has been analyzed very thoroughly by Brinkmann. Whether this is directly relevant to your question is, of course, open to debate.
Peter Brinkmann, MR 1934698 Splittings of mapping tori of free group automorphisms, Geom. Dedicata 93 (2002), 191--203.
