acting freely on ${{\mathbb{R}}}^n$–trees.

After completing this work, the author learnt about the unpublished thesis of Shalom Gross, a student of Z Sela, proving the finite presentation of finitely generated groups having a free action on an ${{\mathbb{R}}}^n$–tree [@Gross_these]. Both proofs deeply rely on Sela’s structure theorem for super-stable actions of finitely generated groups on ${{\mathbb{R}}}$–trees ([@Sela_acylindrical Theorem 3.1], see also Theorem \[thm\_structure\] below). However, Gross does not state a dévissage theorem over cyclic groups, but over finitely generated abelian groups.

Let’s recall briefly the definition of a $\Lambda$–tree. Given a totally ordered abelian group $\Lambda$, there is a natural notion of $\Lambda$–metric space where the distance function takes its values in $\Lambda$. If $\Lambda$ is archimedean, then $\Lambda$ is isomorphic to a subgroup of ${{\mathbb{R}}}$ and we have a metric in the usual sense. When $\Lambda$ is not archimedean, there are elements which are infinitely small compared to other elements. A typical example is when $\Lambda={{\mathbb{R}}}^n$ endowed with the lexicographic ordering.

A $\Lambda$–tree may be defined as a geodesic $0$–hyperbolic $\Lambda$–metric space. Roughly speaking, an ${{\mathbb{R}}}^n$–tree may be thought of as a kind of bundle over an ${{\mathbb{R}}}$–tree where the fibres are (infinitesimal) ${{\mathbb{R}}}^{n-1}$–trees.

In his list of research 