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Vincent Guirardel
Actions of finitely generated groups on $\mathbb{R}$-trees
(Actions de groupes de type fini sur les arbres réels)
Annales de l'institut Fourier, 58 no. 1 (2008), p. 159-211, doi: 10.5802/aif.2348
Article PDF | Reviews MR 2401220 | Zbl pre05267523
Class. Math.: 20E08, 20F65, 20E06
Keywords: R-tree, splitting of group, Rips theory

Résumé - Abstract

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.

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