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.
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.
@article{AIF_2014__64_3_1109_0,
author = {Brieussel, J\'er\'emie},
title = {Folner sets of alternate directed groups},
journal = {Annales de l'Institut Fourier},
volume = {64},
year = {2014},
pages = {1109-1130},
doi = {10.5802/aif.2875},
zbl = {06387302},
mrnumber = {3330165},
language = {en},
url = {http://dml.mathdoc.fr/item/AIF_2014__64_3_1109_0}
}
Brieussel, Jérémie. Folner sets of alternate directed groups. Annales de l'Institut Fourier, Tome 64 (2014) pp. 1109-1130. doi : 10.5802/aif.2875. http://gdmltest.u-ga.fr/item/AIF_2014__64_3_1109_0/
[1] Finite automata and Burnside’s problem for periodic groups, Math. Notes, Tome 11 (1972), pp. 199-203 | Zbl 0253.20049
[2] Amenability of linear-activity automaton groups, J. Eur. Math. Soc. (JEMS), Tome 15 (2013) no. 3, pp. 705-730 | MR 3085088 | Zbl 1277.37019
[3] On amenability of automata groups, Duke Math. J., Tome 154 (2010) no. 3, pp. 575-598 | MR 2730578 | Zbl 1268.20026
[4] Amenability via random walks, Duke Math. J., Tome 130 (2005) no. 1, pp. 39-56 | MR 2176547 | Zbl 1104.43002
[5] Growth behaviors in the range (Preprint arXiv: 1107.1632 to appear in Afrika Matematika)
[6] Amenability and non-uniform growth of some directed automorphism groups of a rooted tree, Math. Z., Tome 263 (2009) no. 2, pp. 265-293 | MR 2534118 | Zbl 1227.43001
[7] Behaviors of entropy on finitely generated groups, Ann. Probab., Tome 41 (2013) no. 6, pp. 4116-4161 | MR 3161471 | Zbl 1280.05123
[8] Isoperimetry for wreath products of Markov chains and multiplicity of selfintersections of random walks, Probab. Theory Related Fields, Tome 136 (2006) no. 4, pp. 560-586 | MR 2257136 | Zbl 1105.60009
[9] On groups with full Banach mean value, Math. Scand., Tome 3 (1955), pp. 243-254 | MR 79220 | Zbl 0067.01203
[10] Degrees of growth of finitely generated groups and the theory of invariant means, Izv. Akad. Nauk SSSR Ser. Mat., Tome 48 (1984) no. 5, pp. 939-985 | MR 764305 | Zbl 0583.20023
[11] Heat kernel upper bounds on a complete non-compact manifold, Rev. Mat. Iberoamericana, Tome 10 (1994) no. 2, pp. 395-452 | Zbl 0810.58040
[12] Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. (1981) no. 53, pp. 53-73 | Numdam | MR 623534 | Zbl 0474.20018
[13] Random walks on discrete groups: boundary and entropy, Ann. Probab., Tome 11 (1983) no. 3, pp. 457-490 | MR 704539 | Zbl 0641.60009
[14] Full Banach mean values on countable groups, Math. Scand., Tome 7 (1959), pp. 146-156 | MR 112053 | Zbl 0092.26704
[15] Some questions of Edjvet and Pride about infinite groups, Illinois J. Math., Tome 30 (1986) no. 2, pp. 301-316 | MR 840129 | Zbl 0598.20029
[16] Croissance des boules et des géodésiques fermées dans les nilvariétés, Ergodic Theory Dynam. Systems, Tome 3 (1983) no. 3, pp. 415-445 | MR 741395 | Zbl 0509.53040
[17] Random walks on finite rank solvable groups, J. Eur. Math. Soc. (JEMS), Tome 5 (2003) no. 4, pp. 313-342 | MR 2017850 | Zbl 1057.20026
[18] Further groups that do not have uniformly exponential growth, J. Algebra, Tome 279 (2004) no. 1, pp. 292-301 | MR 2078400 | Zbl 1133.20034
[19] On exponential growth and uniformly exponential growth for groups, Invent. Math., Tome 155 (2004) no. 2, pp. 287-303 | MR 2031429 | Zbl 1065.20054