that ${{\mathcal {G}}}_n$ denotes the space of groups marked by a generating family of cardinality $n$.

Let ${{\mathcal {T}}}_n\subset{{\mathcal {G}}}_n$ be the set of marked groups having a free action without inversion on some $\Lambda$–tree ($\Lambda$ may vary with the group).

Then ${{\mathcal {T}}}_n$ is closed in ${{\mathcal {G}}}_n$.

We won’t give the proof of this result since this proposition is not sufficient for us as it does not give any control over $\Lambda$. This is why we rather prove the following more technical result.[^1]

For general information of the action of $SL_2(K)$ on its Bruhat–Tits $\Lambda$–tree $BT_K$ where $K$ a field, and $v\co K{\rightarrow}\Lambda\cup\{\infty\}$ is a valuation, see for instance [@Chi_book 4.3, p.144]. Essentially, we will only use the existence of the Bruhat–Tits $\Lambda$–tree and the formula for the translation length of a matrix $m\in SL_2(K)$: $l_{BT_K}(m)=\max\{-2v(\operatorname{Tr}(m)),0\}$. Also note that the action of $SL_2(K)$ on its Bruhat–Tits tree has no inversion (however, there may be inversions in $GL_2(K)$).

By an action of $\Gamma$ on a Bruhat–Tits tree, we mean an action of $\Gamma$ on the Bruhat–Tits $\Lambda$–tree for $SL_2(K)$ induced by a morphism $j\co \Gamma{\rightarrow}SL_2(K)$ where $K$ is a valuated field with values in $\Lambda$.

[thm\_closed]{}

Let ${{{{\mathcal {B}}}{{\mathcal 