Groups of given intermediate word growth

Bartholdi, Laurent; Erschler, Anna

Annales de l'Institut Fourier, Tome 64 (2014) no. 5, pp. 2003-2036.

Nous montrons qu’il existe un groupe de type fini de croissance $\sim f$ pour n’importe quelle fonction $f:{ℝ}_{+}\to {ℝ}_{+}$ satisfaisant $f\left(2R\right)\le f{\left(R\right)}^2\le f\left({\eta }_{+}R\right)$ lorsque $R$ est suffisamment grand, avec ${\eta }_{+}\approx 2.4675$ la racine positive de $X^3-X^2-2X-4$. Soit ${\alpha }_{-}=log2/log{\eta }_{+}\approx 0.7674$  ; alors toutes les fonctions qui croissent uniformément plus vite que $exp\left(R^{{\alpha }_{-}}\right)$ sont réalisables comme fonction de croissance d’un groupe.

Nous exhibons aussi une famille de groupes branchés contractants-pour-la-somme et de croissance $\sim exp\left(R^{\alpha }\right)$, pour un sous-ensemble dense d’$\alpha \in [{\alpha }_{-},1]$.

We show that there exists a finitely generated group of growth $\sim f$ for all functions $f:{ℝ}_{+}\to {ℝ}_{+}$ satisfying $f\left(2R\right)\le f{\left(R\right)}^2\le f\left({\eta }_{+}R\right)$ for all $R$ large enough and ${\eta }_{+}\approx 2.4675$ the positive root of $X^3-X^2-2X-4$. Set ${\alpha }_{-}=log2/log{\eta }_{+}\approx 0.7674$; then all functions that grow uniformly faster than $exp\left(R^{{\alpha }_{-}}\right)$ are realizable as the growth of a group.

We also give a family of sum-contracting branched groups of growth $\sim exp\left(R^{\alpha }\right)$ for a dense set of $\alpha \in [{\alpha }_{-},1]$.

Classification : 20E08, 20F65
Keywords: Growth of groups, self-similar groups, groups acting on trees, wreath products
Mot clés : Croissance des groupes, groupes auto-similaires, groupes agissant sur des arbres, produits en couronne

[1] Bartholdi, Laurent The growth of Grigorchuk’s torsion group, Internat. Math. Res. Notices (1998) no. 20, pp. 1049-1054 | DOI | MR | Zbl

[2] Bartholdi, Laurent Lower bounds on the growth of a group acting on the binary rooted tree, Internat. J. Algebra Comput., Volume 11 (2001) no. 1, pp. 73-88 | DOI | MR | Zbl

[3] Bartholdi, Laurent Endomorphic presentations of branch groups, J. Algebra, Volume 268 (2003) no. 2, pp. 419-443 | DOI | MR | Zbl

[4] Bartholdi, Laurent; Erschler, Anna Growth of permutational extensions, Invent. Math., Volume 189 (2012) no. 2, pp. 431-455 | DOI | MR | Zbl

[5] Bartholdi, Laurent; Erschler, Anna G. Poisson-Furstenberg boundary and growth of groups (arXiv:math/1107.5499)

[6] Bartholdi, Laurent; Grigorchuk, Rostislav I.; Šuniḱ, Zoran Branch groups, Handbook of algebra, Vol. 3, North-Holland, Amsterdam, 2003, pp. 989-1112 | MR | Zbl

[7] Bartholdi, Laurent; Smoktunowicz, Agata Images of Golod-Shafarevich algebras with small growth (to appear in Quartely J. Math, arXiv:math/1108.4267, DOI: 10.1093/qmath/hat005) | MR

[8] Bartholdi, Laurent; Šuniḱ, Zoran On the word and period growth of some groups of tree automorphisms, Comm. Algebra, Volume 29 (2001) no. 11, pp. 4923-4964 | DOI | MR | Zbl

[9] Birkhoff, Garrett Extensions of Jentzsch’s theorem, Trans. Amer. Math. Soc., Volume 85 (1957), pp. 219-227 | MR | Zbl

[10] Brieussel, Jérémie Growth behaviours in the range $e^{\left(r^{\alpha }\right)}$ (arXiv:math/1107.1632)

[11] Brieussel, Jérémie Growth of certain groups of automorphisms of rooted trees, Université de Paris 7 (2008) (Doctoral Dissertation)

[12] Carlitz, L.; Wilansky, A.; Milnor, John; Struble, R. A.; Felsinger, Neal; Simoes, J. M. S.; Power, E. A.; Shafer, R. E.; Maas, R. E. Problems and Solutions: Advanced Problems: 5600-5609, Amer. Math. Monthly, Volume 75 (1968) no. 6, pp. 685-687 | DOI | MR

[13] de Cornulier, Yves Finitely presented wreath products and double coset decompositions, Geom. Dedicata, Volume 122 (2006), pp. 89-108 | DOI | MR | Zbl

[14] Erschler, Anna Boundary behavior for groups of subexponential growth, Ann. of Math. (2), Volume 160 (2004) no. 3, pp. 1183-1210 | DOI | MR | Zbl

[15] Erschler, Anna Critical constants for recurrence of random walks on $G$-spaces, Ann. Inst. Fourier (Grenoble), Volume 55 (2005) no. 2, pp. 493-509 | DOI | Numdam | MR | Zbl

[16] Èrshler, A. G. On the degrees of growth of finitely generated groups, Funktsional. Anal. i Prilozhen., Volume 39 (2005) no. 4, pp. 86-89 | MR | Zbl

[17] Grigorchuk, R. I. On the Milnor problem of group growth, Dokl. Akad. Nauk SSSR, Volume 271 (1983) no. 1, pp. 30-33 | MR | Zbl

[18] 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

[19] Grigorchuk, R. I. Degrees of growth of $p$-groups and torsion-free groups, Mat. Sb. (N.S.), Volume 126(168) (1985) no. 2, p. 194-214, 286 | MR | Zbl

[20] Grigorchuk, R. I.; Machì, A. An example of an indexed language of intermediate growth, Theoret. Comput. Sci., Volume 215 (1999) no. 1-2, pp. 325-327 | DOI | MR | Zbl

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

[22] de la Harpe, Pierre Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000, pp. vi+310 | MR | Zbl

[23] Kassabov, Martin; Pak, Igor Groups of oscillating intermediate growth, Ann. of Math. (2), Volume 177 (2013) no. 3, pp. 1113-1145 | DOI | MR | Zbl

[24] Krause, Günter R.; Lenagan, Thomas H. Growth of algebras and Gelfand-Kirillov dimension, Graduate Studies in Mathematics, 22, American Mathematical Society, Providence, RI, 2000, pp. x+212 | MR | Zbl

[25] Leonov, Yu. G. On a lower bound for the growth function of the Grigorchuk group, Mat. Zametki, Volume 67 (2000) no. 3, pp. 475-477 | MR | Zbl

[26] Mann, Avinoam How groups grow, London Mathematical Society Lecture Note Series, 395, Cambridge University Press, Cambridge, 2012, pp. 1-200 | MR | Zbl

[27] Muchnik, Roman; Pak, Igor On growth of Grigorchuk groups, Internat. J. Algebra Comput. (2001) no. 1, pp. 1-17 | DOI | MR | Zbl

[28] Sidki, Said On a $2$-generated infinite $3$-group: the presentation problem, J. Algebra, Volume 110 (1987) no. 1, pp. 13-23 | DOI | MR | Zbl

[29] Trofimov, V. I. The growth functions of finitely generated semigroups, Semigroup Forum, Volume 21 (1980) no. 4, pp. 351-360 | DOI | MR | Zbl

[30] Warfield, Robert B. Jr. The Gel’fand-Kirillov dimension of a tensor product, Math. Z., Volume 185 (1984) no. 4, pp. 441-447 | DOI | MR | Zbl