group acts freely on an ${{\mathbb{R}}}^n$–tree {#sec_remeslennikov}
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The goal of this section is to prove that limit groups act freely on ${{\mathbb{R}}}^n$–trees. This is an adaptation of an argument by Remeslennikov concerning fully residually free groups ([@Remeslennikov_exist_ukrainian], see also [@Chi_book Theorem 5.5.10 p.246]). Note that it is claimed in [@Remeslennikov_exist_ukrainian] that finitely generated fully residually free groups act freely on a $\Lambda$–tree where $\Lambda$ is a *finitely generated* ordered abelian group. However, the proof is not completely correct since it relies on a misquoted result about valuations (Theorem 3 in [@Remeslennikov_exist_ukrainian]) to which there are known counterexamples (for any subgroup $\Lambda\subset{{\mathbb{Q}}}$, there is valuation on ${{\mathbb{Q}}}(X,Y)$, extending the trivial valuation on ${{\mathbb{Q}}}$, whose value group is $\Lambda$ [@Zariski-Samuel_commutative2 ch.VI, 15, ex.3,4] or [@Kuhlmann_value Theorem 1.1]). Nevertheless, Remeslennikov’s argument proves the following weaker statement: a finitely generated fully residually free group acts freely on a $\Lambda$–tree where $\Lambda$ has finite ${{\mathbb{Q}}}$–rank, [i.e. ]{}$\Lambda{\otimes}{{\mathbb{Q}}}$ is finite dimensional over ${{\mathbb{Q}}}$.

The fact that a limit group acts freely on an ${{\mathbb{R}}}^n$ tree will be deduced from a more general result about group acting freely on Bruhat–Tits trees. But we first state a simpler result in this spirit (see also [@GaSp_does; @GaSp_every]). Remember 