shortening argument. The proof is also much shorter than the one by Kharlampovich–Myasnikov in [@KhMy_irreducible1; @KhMy_irreducible2] using algebraic geometry over groups, and the study of equations in free groups.

The paper is organized as follows: after some premilinaries in section \[sec\_prelim\], section \[sec\_remeslennikov\] is devoted to the proof of the fact that limit groups act freely on ${{\mathbb{R}}}^n$–trees. Section \[sec\_gluing\] sets up some preliminary work on graph of actions on $\Lambda$–trees, which encode how to glue equivariantly some $\Lambda$–trees to get a new $\Lambda$–tree. In section \[sec\_abelian\], starting with a free action of a group $\Gamma$ on an ${{\mathbb{R}}}^n$–tree $T$, we study the action on the ${{\mathbb{R}}}$–tree ${\overline{T}}$ obtained by identifying points at infinitesimal distance, and we deduce a weaker version of the dévissage Theorem where we obtain a graph of groups over (maybe non-finitely generated) abelian groups. Section \[sec\_flawless\] contains the core of the argument: starting with a free action of $\Gamma$ on an ${{\mathbb{R}}}^n$–tree $T$, we build a free action on an ${{\mathbb{R}}}^n$–tree $T'$ such that the ${{\mathbb{R}}}$–tree ${{{\overline{T}}}{}}'$ has *cyclic* arc stabilizers. The dévissage theorem and its corollaries will then follow immediately, as shown in section \[sec\_devissage\].

The author thanks the referee for useful presentation suggestions.

Preliminaries 