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Actions of finitely generated groups on $$\mathbb{R}$$-trees. (English) Zbl 1187.20020
Summary: We study actions of finitely generated groups on $$\mathbb{R}$$-trees under some stability hypotheses. We prove that either the group splits over some controlled subgroup (fixing an arc in particular), or the action can be obtained by gluing together actions of simple types: actions on simplicial trees, actions on lines, and actions coming from measured foliations on 2-orbifolds. This extends results by Sela and Rips-Sela. However, their results are misstated, and we give a counterexample to their statements.
The proof relies on an extended version of Scott’s lemma of independent interest. This statement claims that if a group $$G$$ is a direct limit of groups having suitably compatible splittings, then $$G$$ splits.

##### MSC:
 20E08 Groups acting on trees 20F65 Geometric group theory 20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
##### Keywords:
$$\mathbb{R}$$-trees; splittings of groups; Rips theory
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