which HNN-extensions are free products? this question is related with another still unsolved about Nielsen-Thruston-reducibility and connected-sum-irreducibility of 3d-torus- bundles... 



HNN extensions which are free products
This might help.


**Lemma** If $A$ does not split freely and $C$ is a non-trivial subgroup of $A$ then the HNN extension $G=A\*\_C$ does not split freely.


The proof uses Bass--Serre theory---see Serre's book *Trees* from 1980.


*Proof.* Let $T$ be the Bass--Serre tree of a free splitting of $G$. Because $A$ does not split freely, $A$ stabilizes some unique vertex $v$. But $C$ is non-trivial, so $C$ also stabilizes a unique vertex, which must be $v$. Therefore, $G$ stabilizes $v$, which means the free splitting was trivial. *QED*


A similar argument shows the following.


**Lemma** If $ A\*\_C $ splits non-trivially as an amalgamated free product $ A' \*\_{C'} B'$ then either $A$ splits over $C'$ or $C$ is conjugate into $C'$.


