Roughly speaking, I want to know whether one-relator groups only have 'obvious' free splittings.

> Consider a one-relator group $G=F/\langle\langle r\rangle\rangle$, where $F$ is a free group.  Is it true that $F$ splits non-trivially as a free product $A * B$ if and only if $r$ is contained in a proper free factor of $F$?

**Remarks**

 1. One direction is obvious.  It is clear that if $r$ is contained in a proper free factor then $G$ splits freely.  (We think of $\mathbb{Z}\cong\langle a,b\rangle/\langle\langle b\rangle\rangle$ as an HNN extension of the trivial group, so it's not really a counterexample, even though it might look like one.)
 2. A quick search of the literature suggests that the isomorphism problem for one-relator groups is wide open.  (I'd be interested in any details that anyone may have.)
 3. There is no decision-theoretic obstruction.  Magnus famously solved the word problem for one-relator groups. Much more recently, [Nicholas Touikan has shown][1] that, for any finitely generated group, if you can solve the word problem then you can compute the Grushko decomposition.  So one can algorithmically determine whether a given one-relator group splits.  If the answer to my question is 'yes' then one can use Whitehead's Algorithm to find this out comparatively quickly.
 4. When I first considered this question, it seemed to me that the answer was obviously 'yes' - I don't see how there could possibly be room in a presentation 2-complex for a 'non-obvious' free splitting.  But a proof has eluded me, and of course many seemingly obvious facts about one-relator groups are extremely hard to prove.


  [1]: http://arxiv.org/abs/0906.3902