Actions of finitely generated groups on $ℝ$-trees

Guirardel, Vincent

Annales de l'Institut Fourier, Tome 58 (2008) no. 1, pp. 159-211.

On étudie les actions de groupes de type fini sur des arbres réels sous certaines hypothèses de stabilité. On démontre que soit le groupe se scinde au dessus de sous-groupes contrôlés (fixant un arc en particulier), soit que l’action peut être obtenue par recollement d’actions simples : actions sur des arbres simpliciaux, actions sur des droites, et actions venant de feuilletages mesurés sur des $2$-orbifolds. Ceci étend des résultats de Sela et de Rips-Sela. Cependant, leurs résultats sont mal énoncés, et on donne un contrexemple à leurs énoncés.

La preuve repose sur une version étendue du Lemme de Scott qui est intéressante en soi. Cet énoncé affirme que si un groupe $G$ est une limite directe de groupes ayant des scindements compatibles en un sens convenable, alors $G$ se scinde.

We study actions of finitely generated groups on $ℝ$-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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