Bartholdi, Laurent; Erschler, Anna
Groups of given intermediate word growth  [ Groupes de croissance intermédiaire donnée ]
Annales de l'institut Fourier, Tome 64 (2014) no. 5 , p. 2003-2036
MR 3330929 | Zbl 06387329
doi : 10.5802/aif.2902
URL stable : http://www.numdam.org/item?id=AIF_2014__64_5_2003_0

Classification:  20E08,  20F65
Mots clés: Croissance des groupes, groupes auto-similaires, groupes agissant sur des arbres, produits en couronne
Nous montrons qu’il existe un groupe de type fini de croissance f pour n’importe quelle fonction f: + + satisfaisant f(2R)f(R) 2 f(η + R) lorsque R est suffisamment grand, avec η + 2.4675 la racine positive de X 3 -X 2 -2X-4. Soit α - =log2/logη + 0.7674  ; alors toutes les fonctions qui croissent uniformément plus vite que exp(R α - ) sont réalisables comme fonction de croissance d’un groupe. Nous exhibons aussi une famille de groupes branchés contractants-pour-la-somme et de croissance exp(R α ), pour un sous-ensemble dense d’α[α - ,1].
We show that there exists a finitely generated group of growth f for all functions f: + + satisfying f(2R)f(R) 2 f(η + R) for all R large enough and η + 2.4675 the positive root of X 3 -X 2 -2X-4. Set α - =log2/logη + 0.7674; then all functions that grow uniformly faster than exp(R α - ) are realizable as the growth of a group. We also give a family of sum-contracting branched groups of growth exp(R α ) for a dense set of α[α - ,1].

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