Folner sets of alternate directed groups
Annales de l'Institut Fourier, Volume 64 (2014) no. 3, pp. 1109-1130.

An explicit family of Folner sets is constructed for some directed groups acting on a rooted tree of sublogarithmic valency by alternate permutations. In the case of bounded valency, these groups were known to be amenable by probabilistic methods. The present construction provides a new and independent proof of amenability, using neither random walks, nor word length.

On construit une famille explicite d’ensembles de Folner pour certains groupes dirigés agissant sur des arbres enracinés à valence sous-logarithmique par des permutations alternées. Dans le cas d’arbres à valence bornée, la moyennabilité de ces groupes avait déjà été prouvée au moyen de techniques probabilistes. La construction présentée ici fournit une nouvelle preuve, n’utilisant ni marches aléatoires, ni longueur des mots.

DOI: 10.5802/aif.2875
Classification: 20E08, 20F65, 43A00
Keywords: Groups acting on rooted trees, directed groups, bounded automata groups, Folner sets, amenability
Mot clés : groupes agissant sur des arbres enracinés, groupes dirigés, groupes d’automates bornés, ensembles de Folner, moyennabilité

Brieussel, Jérémie 1

1 Université Montpellier 2 Place E. Bataillon 34095 Montpellier (France)
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Brieussel, Jérémie. Folner sets of alternate directed groups. Annales de l'Institut Fourier, Volume 64 (2014) no. 3, pp. 1109-1130. doi : 10.5802/aif.2875. https://aif.centre-mersenne.org/articles/10.5802/aif.2875/

[1] Aleshin, S. Finite automata and Burnside’s problem for periodic groups, Math. Notes, Volume 11 (1972), pp. 199-203 | Zbl

[2] Amir, Gideon; Angel, Omer; Virág, Bálint Amenability of linear-activity automaton groups, J. Eur. Math. Soc. (JEMS), Volume 15 (2013) no. 3, pp. 705-730 | MR | Zbl

[3] Bartholdi, Laurent; Kaimanovich, Vadim A.; Nekrashevych, Volodymyr V. On amenability of automata groups, Duke Math. J., Volume 154 (2010) no. 3, pp. 575-598 | MR | Zbl

[4] Bartholdi, Laurent; Virág, Bálint Amenability via random walks, Duke Math. J., Volume 130 (2005) no. 1, pp. 39-56 | MR | Zbl

[5] Brieussel, Jérémie Growth behaviors in the range e r α (Preprint arXiv: 1107.1632 to appear in Afrika Matematika)

[6] Brieussel, Jérémie Amenability and non-uniform growth of some directed automorphism groups of a rooted tree, Math. Z., Volume 263 (2009) no. 2, pp. 265-293 | MR | Zbl

[7] Brieussel, Jérémie Behaviors of entropy on finitely generated groups, Ann. Probab., Volume 41 (2013) no. 6, pp. 4116-4161 | MR | Zbl

[8] Erschler, Anna Isoperimetry for wreath products of Markov chains and multiplicity of selfintersections of random walks, Probab. Theory Related Fields, Volume 136 (2006) no. 4, pp. 560-586 | MR | Zbl

[9] Følner, Erling On groups with full Banach mean value, Math. Scand., Volume 3 (1955), pp. 243-254 | MR | Zbl

[10] Grigorchuk, R. I. Degrees of growth of finitely generated groups and the theory of invariant means, Izv. Akad. Nauk SSSR Ser. Mat., Volume 48 (1984) no. 5, pp. 939-985 | MR | Zbl

[11] Grigorʼyan, Alexander Heat kernel upper bounds on a complete non-compact manifold, Rev. Mat. Iberoamericana, Volume 10 (1994) no. 2, pp. 395-452 | Zbl

[12] Gromov, Mikhael Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. (1981) no. 53, pp. 53-73 | Numdam | MR | Zbl

[13] Kaĭmanovich, V. A.; Vershik, A. M. Random walks on discrete groups: boundary and entropy, Ann. Probab., Volume 11 (1983) no. 3, pp. 457-490 | MR | Zbl

[14] Kesten, Harry Full Banach mean values on countable groups, Math. Scand., Volume 7 (1959), pp. 146-156 | MR | Zbl

[15] Neumann, Peter M. Some questions of Edjvet and Pride about infinite groups, Illinois J. Math., Volume 30 (1986) no. 2, pp. 301-316 | MR | Zbl

[16] Pansu, Pierre Croissance des boules et des géodésiques fermées dans les nilvariétés, Ergodic Theory Dynam. Systems, Volume 3 (1983) no. 3, pp. 415-445 | MR | Zbl

[17] Pittet, Ch.; Saloff-Coste, L. Random walks on finite rank solvable groups, J. Eur. Math. Soc. (JEMS), Volume 5 (2003) no. 4, pp. 313-342 | MR | Zbl

[18] Wilson, John S. Further groups that do not have uniformly exponential growth, J. Algebra, Volume 279 (2004) no. 1, pp. 292-301 | MR | Zbl

[19] Wilson, John S. On exponential growth and uniformly exponential growth for groups, Invent. Math., Volume 155 (2004) no. 2, pp. 287-303 | MR | Zbl

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