Fima, Pierre
Amenable, transitive and faithful actions of groups acting on trees  [ Actions moyennables, transitives et fidèles de groupes agissant sur un arbre ]
Annales de l'institut Fourier, Tome 64 (2014) no. 1 , p. 1-17
MR 3330539 | Zbl 06387264
doi : 10.5802/aif.2837
URL stable : http://www.numdam.org/item?id=AIF_2014__64_1_1_0

Classification:  43A07,  20E06,  57M07
Mots clés: actions moyennable, produit libre, extension HNN, groupes agissant sur un arbre
Nous étudions sous quelles conditions un produit libre amalgamé ou une extension HNN sur un sous groupe fini admet une action moyennable, transitive et fidèle sur un espace dénombrable. Nous montrons qu’une telle action existe lorsque les groupes initiaux admettent une action moyennable et presque libre à orbites infinies (e.g. les groupes virtuellement libres ou moyennables infinis). Notre résultat s’appuie sur le théorème de Baire. Nous étendons ce résultat aux groupes agissant sur un arbre.
We study under which condition an amalgamated free product or an HNN-extension over a finite subgroup admits an amenable, transitive and faithful action on an infinite countable set. We show that such an action exists if the initial groups admit an amenable and almost free action with infinite orbits (e.g. virtually free groups or infinite amenable groups). Our result relies on the Baire category Theorem. We extend the result to groups acting on trees.

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