Amenable, transitive and faithful actions of groups acting on trees
Annales de l'Institut Fourier, Volume 64 (2014) no. 1, pp. 1-17.

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.

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.

DOI: 10.5802/aif.2837
Classification: 43A07, 20E06, 57M07
Keywords: amenable action, free product, HNN extension, groups acting on trees
Mot clés : actions moyennable, produit libre, extension HNN, groupes agissant sur un arbre
Fima, Pierre 1

1 Université Denis-Diderot Paris 7, IMJ, Bâtiment Sophie Germain, case 7012, 75205 Paris cedex 13
@article{AIF_2014__64_1_1_0,
     author = {Fima, Pierre},
     title = {Amenable, transitive and faithful actions of groups acting on trees},
     journal = {Annales de l'Institut Fourier},
     pages = {1--17},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {64},
     number = {1},
     year = {2014},
     doi = {10.5802/aif.2837},
     zbl = {1315.43001},
     mrnumber = {3330539},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2837/}
}
TY  - JOUR
AU  - Fima, Pierre
TI  - Amenable, transitive and faithful actions of groups acting on trees
JO  - Annales de l'Institut Fourier
PY  - 2014
SP  - 1
EP  - 17
VL  - 64
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2837/
DO  - 10.5802/aif.2837
LA  - en
ID  - AIF_2014__64_1_1_0
ER  - 
%0 Journal Article
%A Fima, Pierre
%T Amenable, transitive and faithful actions of groups acting on trees
%J Annales de l'Institut Fourier
%D 2014
%P 1-17
%V 64
%N 1
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2837/
%R 10.5802/aif.2837
%G en
%F AIF_2014__64_1_1_0
Fima, Pierre. Amenable, transitive and faithful actions of groups acting on trees. Annales de l'Institut Fourier, Volume 64 (2014) no. 1, pp. 1-17. doi : 10.5802/aif.2837. https://aif.centre-mersenne.org/articles/10.5802/aif.2837/

[1] Dixon, J. D. Most finitely generated permutation groups are free, Bull. London Math. Soc., Volume 22 (1990) no. 3, pp. 222-226 | DOI | MR | Zbl

[2] Epstein, D. B. A. Almost all subgroups of a Lie group are free, J. Algebra, Volume 19 (1971), pp. 261-262 | DOI | MR | Zbl

[3] Glasner, Y.; Monod, N. Amenable action, free products and a fixed point property, Bull. Lond. Math. Soc., Volume 39 (2007) no. 1, pp. 138-150 | DOI | MR | Zbl

[4] Greenleaf, F. P. Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, 16, Van Nostrand Reinhold Co., New York, 1969 | MR | Zbl

[5] Grigorchuk, R.; Nekrashevych, V. Amenable actions of non amenable groups, Zap. Nauchn. Sem. S.-Peterburg Otdel. Math. Inst. Steklov (POMI), Volume 326 (2005), pp. 85-96 | MR | Zbl

[6] Monod, N.; Popa, S. On co-amenability for groups and von Neumann algebras, C. R. Math. Acad. Sci. Soc. R. Can., Volume 25 (2003) no. 3, pp. 82-87 | MR | Zbl

[7] Moon, S. Amenable actions of amalgamated free products, Groups, Geometry and Dynamics, Volume 4 (2010) no. 2, pp. 309-332 | DOI | MR | Zbl

[8] Moon, S. Amenable actions of amalgamated free products of free groups over a cyclic subgroup and generic property, Ann. Math. Blaise Pascal, Volume 18 (2011) no. 2, pp. 217-235 | DOI | Numdam | MR | Zbl

[9] Moon, S. Permanent properties of amenable, transitive and faithful actions, Bull. Belgian Math. Soc. Simon Stevin, Volume 18 (2011) no. 2, pp. 287-296 | MR | Zbl

[10] Serre, J.-P. Arbres, amalgames, SL 2 , Astérisque, 46, 1983 | Zbl

[11] van Douwen, E. K. Measures invariant under actions of 𝔽 2 , Topology Appl., Volume 34 (1990) no. 1, pp. 53-68 | DOI | MR | Zbl

[12] von Neumann, J. Zusatz zur Arbeit “Zur allgemeinen Theorie des Masses”, Fund. Math., Volume 13 (1929), pp. 73-116

[13] Zimmer, R. J. Ergodic theory and semisimple groups,, Birkhäuser, Basel, 1984 | MR | Zbl

Cited by Sources: