no. 5
Groups of given intermediate word growth
[Groupes de croissance intermédiaire donnée]
Bartholdi, Laurent1 ; Erschler, Anna2
1 Georg-August Universität L.B.: Mathematisches Institut Bunsenstraße 3–5 D-37073 Göttingen (Germany)
2 A.E.: C.N.R.S. Université Paris Sud Département de Mathématiques Bâtiment 425 91405 Orsay (France)
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 ∼f pour n’importe quelle fonction f:ℝ + →ℝ + 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:ℝ + →ℝ + 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].

DOI : 10.5802/aif.2902
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

Bartholdi, Laurent 1 ; Erschler, Anna 2

1 Georg-August Universität L.B.: Mathematisches Institut Bunsenstraße 3–5 D-37073 Göttingen (Germany)
2 A.E.: C.N.R.S. Université Paris Sud Département de Mathématiques Bâtiment 425 91405 Orsay (France) 
@article{AIF_2014__64_5_2003_0,
     author = {Bartholdi, Laurent and Erschler, Anna},
     title = {Groups of given intermediate word growth},
     journal = {Annales de l'Institut Fourier},
     pages = {2003--2036},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {64},
     number = {5},
     year = {2014},
     doi = {10.5802/aif.2902},
     mrnumber = {3330929},
     zbl = {06387329},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2902/}
}
TY  - JOUR
AU  - Bartholdi, Laurent
AU  - Erschler, Anna
TI  - Groups of given intermediate word growth
JO  - Annales de l'Institut Fourier
PY  - 2014
SP  - 2003
EP  - 2036
VL  - 64
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2902/
DO  - 10.5802/aif.2902
LA  - en
ID  - AIF_2014__64_5_2003_0
ER  - 
%0 Journal Article
%A Bartholdi, Laurent
%A Erschler, Anna
%T Groups of given intermediate word growth
%J Annales de l'Institut Fourier
%D 2014
%P 2003-2036
%V 64
%N 5
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2902/
%R 10.5802/aif.2902
%G en
%F AIF_2014__64_5_2003_0
Bartholdi, Laurent; Erschler, Anna. Groups of given intermediate word growth. Annales de l'Institut Fourier, Tome 64 (2014) no. 5, pp. 2003-2036. doi : 10.5802/aif.2902. https://aif.centre-mersenne.org/articles/10.5802/aif.2902/

[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 (r α ) (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

Cité par Sources :