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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?

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  • $\begingroup$ Why do you think this might be true? $\endgroup$ – Igor Rivin Sep 7 '16 at 16:27
  • $\begingroup$ @IgorRivin I have seen this in some examples, and to be honest, I do not have enough evidence to think that this should hold. I would of course be happy to see a counterexample. $\endgroup$ – Pablo Sep 7 '16 at 16:28
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    $\begingroup$ If $\tau$ is inner then $G$ is isomorphic to $F\times\mathbf{Z}$. This is an amalgam (write $F=F'*F''$, then $G$ is the amalgam $(F'\times\mathbf{Z})*_{(\{1\}\times\mathbf{Z})}(F''\times\mathbf{Z})$. $\endgroup$ – YCor Sep 8 '16 at 11:40
  • $\begingroup$ @YCor You are right, I missed that. I just wonder what is a simple example for an automorphism of $F$ that is not atoroidal. $\endgroup$ – Pablo Sep 8 '16 at 11:45
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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.

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  • $\begingroup$ There seems also to be a hyperbolicity assumption on $G$. $\endgroup$ – Pablo Sep 8 '16 at 3:48
  • $\begingroup$ the atoroidal assumption implies that the extension by Z is word-hyperbolic by a result of Bestvina and Handel. $\endgroup$ – Ian Agol Sep 8 '16 at 3:51
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    $\begingroup$ And why does the existence of a free and cocompact action on a CAT(0) cube complex implies a splitting as an amalgamated product? $\endgroup$ – Pablo Sep 8 '16 at 3:56
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    $\begingroup$ Another way to deduce that it's virtually an AFP: By Hagen--Wise, it's cubulayed. By Agol, it's virtually special. By Ping--Pong, it contains a quasiconvex free subgroup F. By Haglund--Wise, F is a retract of some finite-index subgroup U. Since F splits as an AFP, it follows immediately that U does too. $\endgroup$ – HJRW Sep 10 '16 at 16:18
  • $\begingroup$ One can simplify the above arguments via the following general fact: if a group maps onto a non-trivial free product then it splits as an AFP over the kernel of this homomorphism. By Hagen-Wise and Haglund-Wise the group, in the atoroidal case, virtually embeds into a RAAG. It s well-known that non-abelian subgroups of RAAGs are large, so they always have a f.i. subgroup mapping onto the free product $\mathbb{Z}*\mathbb{Z}$. $\endgroup$ – Ashot Minasyan Sep 13 '16 at 9:50
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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.

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    $\begingroup$ Brinkmann's paper actually discusses splittings over ${\mathbb Z}$ only. $\endgroup$ – ThiKu Sep 8 '16 at 9:05

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