Groups of given intermediate word growth

Bartholdi, Laurent; Erschler, Anna

Groups of given intermediate word growth
[Groupes de croissance intermédiaire donnée]

Annales de l'Institut Fourier, Tome 64 (2014), p. 2003-2036 / Harvested from Numdam

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Abstract

Nous montrons qu’il existe un groupe de type fini de croissance $\sim f$ pour n’importe quelle fonction $f : ℝ_{+} \to ℝ_{+}$ satisfaisant $f (2 R) \leq f {(R)}^{2} \leq f (η_{+} R)$ lorsque $R$ est suffisamment grand, avec $η_{+} \approx 2.4675$ la racine positive de $X^{3} - X^{2} - 2 X - 4$ . Soit $α_{-} = log 2 / log η_{+} \approx 0.7674$ ; 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 $\sim exp (R^{α})$ , pour un sous-ensemble dense d’ $α \in [α_{-}, 1]$ .

We show that there exists a finitely generated group of growth $\sim f$ for all functions $f : ℝ_{+} \to ℝ_{+}$ satisfying $f (2 R) \leq f {(R)}^{2} \leq f (η_{+} R)$ for all $R$ large enough and $η_{+} \approx 2.4675$ the positive root of $X^{3} - X^{2} - 2 X - 4$ . Set $α_{-} = log 2 / log η_{+} \approx 0.7674$ ; 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 $\sim exp (R^{α})$ for a dense set of $α \in [α_{-}, 1]$ .

Publié le : 2014-01-01

MR 3330929 | Zbl 06387329

DOI : https://doi.org/10.5802/aif.2902
Classification: 20E08, 20F65
Mots clés: Croissance des groupes, groupes auto-similaires, groupes agissant sur des arbres, produits en couronne

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     author = {Bartholdi, Laurent and Erschler, Anna},
     title = {Groups of given intermediate word growth},
     journal = {Annales de l'Institut Fourier},
     volume = {64},
     year = {2014},
     pages = {2003-2036},
     doi = {10.5802/aif.2902},
     zbl = {06387329},
     mrnumber = {3330929},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2014__64_5_2003_0}
}

Bartholdi, Laurent; Erschler, Anna. Groups of given intermediate word growth. Annales de l'Institut Fourier, Tome 64 (2014) pp. 2003-2036. doi : 10.5802/aif.2902. http://gdmltest.u-ga.fr/item/AIF_2014__64_5_2003_0/

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