problems, Sela conjectures that a finitely generated group is a limit group if and only if it acts freely on an ${{\mathbb{R}}}^n$–tree [@Sela_problems]. However, it is known that the fundamental group $\Gamma=\langle a,b,c\,|\, a^2b^2c^2=1\rangle$ of the non-orientable surface $\Sigma$ of Euler characteristic $-1$ is not a limit group since three elements in a free group satisfying $a^2b^2c^2=1$ must commute ([@Lyndon_equation],[@Chi_book p.249]). But this group acts freely on a ${{\mathbb{Z}}}^2$–tree: $\Sigma$ can be obtained by gluing together the two boundary components of a twice punctured projective plane, so $\Gamma$ can be written as an HNN extension $F_2*_{{\mathbb{Z}}}$. The ${{\mathbb{Z}}}^2$–tree can be roughly described as the Bass–Serre tree of this HNN extension, but where one blows up each vertex into an infinitesimal tree corresponding to a Cayley graph of $F_2$ (see [@Chi_book p.237] for details).

In this paper, we start by giving a proof that every limit group acts freely on an ${{\mathbb{R}}}^n$–tree. This is an adaptation a theorem by Remeslennikov saying that a fully residually free groups act freely on a $\Lambda$–tree where $\Lambda$ has finite ${{\mathbb{Q}}}$–rank, [i.e. ]{}$\Lambda{\otimes}{{\mathbb{Q}}}$ is finite dimensional ([@Remeslennikov_exist_ukrainian], see also [@Chi_book Theorem 5.10]). However, Remeslennikov claims that $\Lambda$ can be chosen finitely generated, but this relies 