with the union of the translation axes of $\Gamma$; finally, there is no unique such minimal subtree for the action of $\Gamma=\{0\}\times {{\mathbb{Z}}}$ on ${{\mathbb{R}}}^2$.

Elementary properties of groups acting freely on $\Lambda$–trees
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We now recall some elementary properties of groups acting freely (without inversion) on $\Lambda$–trees. They are proved for instance in [@Chi_book].

\[lem\_elementary\] Let $\Gamma$ be a group acting freely without inversion on a $\Lambda$–tree. Then

$\Gamma$ is torsion free;

two elements $g,h\in\Gamma$ commute if and only if they have the same axis. If they don’t commute, the intersection of their axes is either empty or a segment ([@Chi_book], proof of Lemma 5.1.2 p.218 and Rk p.111);

maximal abelian subgroups of $\Gamma$ are malnormal (property CSA) and $\Gamma$ is commutative transitive: the relation of commutation on $\Gamma\setminus \{1\}$ is transitive [@Chi_book Lemma 5.1.2 p.218].

Property CSA implies that $\Gamma$ is commutative transitive.

A result known as *Harrison Theorem*, proved by Harrison for ${{\mathbb{R}}}$–trees and by Chiswell and Urbanski–Zamboni for general $\Lambda$–trees, says that a $2$–generated group acting freely without inversion on a $\Lambda$–tree is either a free group or a free abelian group. (see [@Chi_harrison; @Urbanski-Zamboni_Free; @Harrison_real]). We won’t use this result in this paper.

A limit 