For each $(G,S)\in{{\mathcal {G}}}_n$ and each $R>0$, $N_R(G,S)$ is the set of marked groups $(G',S')\in{{\mathcal {G}}}_n$ such that $(G,S)$ and $(G',S')$ have exactly the same relations of length at most $R$. For this topology, ${{\mathcal {G}}}_n$ is a Hausdorff, compact, totally disconnected space.

A *limit group* $(G,S)\in{{\mathcal {G}}}_n$ is a marked group which is a limit of markings of free groups in ${{\mathcal {G}}}_n$.

Actually, being a limit group does not depend on the choice of the generating set. Moreover, limit groups have several equivalent characterizations: a finitely generated group is a limit group if and only if it is fully residually free, if and only if it has the same universal theory as a free group, if and only if it is a subgroup of a non-standard free group [@Remeslennikov_exist_siberian; @CG_compactifying]. We won’t need those characterizations in this paper.

$\Lambda$–trees
---------------

For background on $\Lambda$–trees, see [@Bass_non-archimedean; @Chi_book].

#### Totally ordered abelian groups

A totally ordered abelian group $\Lambda$ is an abelian group with a total ordering such that for all $x,y,z\in \Lambda$, $x\leq y \Rightarrow x+z\leq y+z$. Our favorite example will be ${{\mathbb{R}}}^n$, with the lexicographic ordering. In all this paper, ${{\mathbb{R}}}^n$ will always be endowed with its 