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In the article Limit groups and groups acting freely on $\mathbb{R}^n$-trees by Vincent Guirardel, after the statement of Theorem 7.1 on page 1430, the author concludes: "Hence, a limit group can be obtained from abelian and surface groups by a finite sequence of free products and amalgamations over Z".

Question: Why are HNN-extensions not necessary in this decomposition?

Here is the full reference to the cited article:
Vincent Guirardel, Limit groups and groups acting freely on $\mathbb{R}^n$-trees, Geometry & Topology 8 (2004), pages 1427-1470
DOI: 10.2140/gt.2004.8.1427, arXiv: math/0306306 [math.GR]

I would need a clarification about a statement in the article Limit groups and groups acting freely on $\mathbb{R}^n$-trees by Vincent Guirardel.

First recall that a limit group is a finitely generated group that is a limit of free groups in the space of marked groups (regardless of a choice of marking). This is (non-trivially) equivalent to: a finitely generated group that is fully residually free. Such groups are automatically finitely presented. (And they are obviously torsion-free, so that in the next few lines "cyclic" means "either trivial or infinite cyclic".)

Here are now the relevant excerpts. First, Corollary 3.4 (p1429), with reference to Remeslennikov: Any limit group has a free action on an $\mathbf{R}^n$-tree [for some $n$].

Then, Theorem 7.1 (p1430): "dévissage theorem": if a finitely generated, freely indecomposable group has a free action on an $\mathbf{R}^n$-tree ($n\ge 2$), then it decomposes as the Bass-Serre fundamental group of a finite graph of groups, with cyclic edge groups and each of whose vertex groups admits a free action on an $\mathbf{R}^{n-1}$-tree.

Just after the statement of this theorem, the author recalls Rips' result that freely indecomposable finitely generated groups with a free action on an $\mathbf{R}$-tree are either free abelian or fundamental groups of closed surfaces (of negative curvature).

Then he deduces Hence, a limit group can be obtained from abelian and surface groups by a finite sequence of free products and amalgamations over $\mathbf{Z}$.

Question: Are HNN-extensions (with cyclic edge groups) really unnecessary in this decomposition?

Here is the full reference to the cited article:
Vincent Guirardel, Limit groups and groups acting freely on $\mathbb{R}^n$-trees, Geometry & Topology 8 (2004), pages 1427-1470
DOI: 10.2140/gt.2004.8.1427, arXiv: math/0306306 [math.GR]

In the article Limit groups and groups acting freely on $\mathbb{R}^n$-trees by Vincent Guirardel, after the statement of Theorem 7.1 in page 1430, the author concludes: "Hence, a limit group can be obtained from abelian and surface groups by a finite sequence of free products and amalgamations over Z".

Question: Why are HNN-extensions not necessary in this decomposition?

Here is the full reference to the cited article:
Vincent Guirardel, Limit groups and groups acting freely on $\mathbb{R}^n$-trees, Geometry & Topology 8 (2004), pages 1427-1470
DOI: 10.2140/gt.2004.8.1427, arXiv:math/0306306 [math.GR]

In the article Limit groups and groups acting freely on $\mathbb{R}^n$-trees by Vincent Guirardel, after the statement of Theorem 7.1 on page 1430, the author concludes: "Hence, a limit group can be obtained from abelian and surface groups by a finite sequence of free products and amalgamations over Z".

Question: Why are HNN-extensions not necessary in this decomposition?

Here is the full reference to the cited article:
Vincent Guirardel, Limit groups and groups acting freely on $\mathbb{R}^n$-trees, Geometry & Topology 8 (2004), pages 1427-1470
DOI: 10.2140/gt.2004.8.1427, arXiv: math/0306306 [math.GR]

In the article Limit groups and groups acting freely on $\mathbb{R}^n$-trees by Vincent Guirardel, after the statement of Theorem 7.1 in page 1430, the author concludes: "Hence, a limit group can be obtained from abelian and surface groups by a finite sequence of free products and amalgamations over Z".

Question: Why are HNN-extensions not necessary in this decomposition?

Here is the full reference to the cited article:
Vincent Guirardel, Limit groups and groups acting freely on $\mathbb{R}^n$-trees, Geometry & Topology 8 (2004), pages 1427-1470, DOI: 10.2140/gt.2004.8.1427, arXiv:math/0306306 [math.GR]

In the article Limit groups and groups acting freely on $\mathbb{R}^n$-trees by Vincent Guirardel, after the statement of Theorem 7.1 in page 1430, the author concludes: "Hence, a limit group can be obtained from abelian and surface groups by a finite sequence of free products and amalgamations over Z".

Question: Why are HNN-extensions not necessary in this decomposition?

Here is the full reference to the cited article:
Vincent Guirardel, Limit groups and groups acting freely on $\mathbb{R}^n$-trees, Geometry & Topology 8 (2004), pages 1427-1470
DOI: 10.2140/gt.2004.8.1427, arXiv:math/0306306 [math.GR]