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...
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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. 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 $C'$. |
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Let me add an explicit partial solution to the question above: for a torus bundle $E$ over the circle, the fundamental group of $E$ can't be a free product of groups, because if it were, the fundamental subgroup of the fiber would be a free product (by the Kurosch's theorem) which is impossible for the torus, then $E$ isn't a connected sum, hence irreducible. At least the HNN extensions which are free products can't be torus bundles, and in fact, no other surface bundles unless the surface be the 2-sphere |
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