on a misquoted result about valuations (see section \[sec\_remeslennikov\]).

We actually prove that there is a closed subspace of the space of marked groups consisting of groups acting freely on ${{\mathbb{R}}}^n$–trees. In the following statement, an action of a group $\Gamma$ on a Bruhat–Tits tree is the action on the Bruhat–Tits tree of $SL_2(K)$ induced by a morphism $j\co \Gamma{\rightarrow}SL_2(K)$ where $K$ is a valuation field. Note that $K$ may vary with $\Gamma$.

[thm\_closed]{}

Let ${{{{\mathcal {B}}}{{\mathcal {T}}}}}\subset{{\mathcal {G}}}_n$ be the set of marked groups $(\Gamma,S)$ having a free action on a Bruhat–Tits tree.

Then ${{{{\mathcal {B}}}{{\mathcal {T}}}}}$ is closed in ${{\mathcal {G}}}_n$, and ${{{{\mathcal {B}}}{{\mathcal {T}}}}}$ consists in groups acting freely on $\Lambda$–trees where $\Lambda{\otimes}{{\mathbb{Q}}}$ has dimension at most $3n+1$ over ${{\mathbb{Q}}}$. In particular, ${{{{\mathcal {B}}}{{\mathcal {T}}}}}$ consists in groups acting freely on ${{\mathbb{R}}}^{3n+1}$–trees where ${{\mathbb{R}}}^{3n+1}$ has the lexicographic ordering.

[Remeslennikov]{}

[[@Remeslennikov_exist_ukrainian]]{}A limit group has a free action on an ${{\mathbb{R}}}^n$–tree.

As a corollary of their study of the structure of limit groups [@KhMy_irreducible1; @KhMy_irreducible2], Kharlampovich and Myasnikov prove the more precise result that a limit group is a subgroup of an iterated free extension of centralizers of a free group, and has therefore a free action on a 