${{\mathbb{Z}}}^n$–tree [@KhMy_irreducible2 Corollary 6]. An alternative proof of this fact using Sela’s techniques is given in [@CG_compactifying].

The main result of the paper is the following structure theorem for groups acting freely on ${{\mathbb{R}}}^n$–trees (see theorem \[cyclic\_devissage\] for a more detailed version). In view of the previous corollary, this theorem applies to limit groups.

[devissage\_simple]{}

Consider a finitely generated, freely indecomposable group $\Gamma$ having a free action on an ${{\mathbb{R}}}^n$–tree. Then $\Gamma$ can be written as the fundamental group of a finite graph of groups with cyclic edge groups and where each vertex group is finitely generated and has a free action on an ${{\mathbb{R}}}^{n-1}$–tree.

For $n=1$, Rips theorem says that $\Gamma$ (which is supposed to be freely indecomposable) is either a free abelian group, or a surface group (see [@GLP1; @BF_stable]). Hence, a limit group can be obtained from abelian and surface groups by a finite sequence of free products and amalgamations over ${{\mathbb{Z}}}$. It is therefore easy to deduce the following result:

[cor\_FP]{}

Let $\Gamma$ be a finitely generated group having a free action on an ${{\mathbb{R}}}^n$–tree. Then

$\Gamma$ is finitely presented [@Gross_these Corollary 6.6];

if $\Gamma$ is not cyclic, then its first Betti number is at least 