Geodesic
Product set growth in Burnside groups
[Croissance des ensembles produit dans les groupes de Burnside]
Coulon, Rémi1 ; Steenbock, Markus2
1 IRMAR, Univ Rennes et CNRS 35000 Rennes, France
2 Fakultät für Mathematik, Universität Wien 1090 Wien, Austria
Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 463-504.

Voir la notice de l'article provenant de la source Numdam

Given a periodic quotient of a torsion-free hyperbolic group, we provide a fine lower estimate of the growth function of any sub-semi-group. This generalizes results of Razborov and Safin for free groups.

Étant donné un quotient périodique d’un groupe hyperbolique sans torsion, nous donnons une estimation inférieure fine de la fonction de croissance pour chacun de tous ses sous-semi-groupes. Cet énoncé généralise des résultats de Razborov et Safin pour les groupes libres.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/jep.187
Classification : 20F65, 20F67, 20F50, 20F06, 20F69
Keywords: Product sets, growth, hyperbolic groups, acylindrical actions, small cancellation, infinite periodic groups, Burnside problem
Mots-clés : Ensemble produit, croissance, groupes hyperboliques, actions cylindriques, théorie de la petite simplification, groupes périodiques infinis, problème de Burnside

Coulon, Rémi 1 ; Steenbock, Markus 2

1 IRMAR, Univ Rennes et CNRS 35000 Rennes, France
2 Fakultät für Mathematik, Universität Wien 1090 Wien, Austria
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{JEP_2022__9__463_0,
     author = {Coulon, R\'emi and Steenbock, Markus},
     title = {Product set growth in {Burnside} groups},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {463--504},
     publisher = {\'Ecole polytechnique},
     volume = {9},
     year = {2022},
     doi = {10.5802/jep.187},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.5802/jep.187/}
}
TY  - JOUR
AU  - Coulon, Rémi
AU  - Steenbock, Markus
TI  - Product set growth in Burnside groups
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2022
SP  - 463
EP  - 504
VL  - 9
PB  - École polytechnique
UR  - http://geodesic.mathdoc.fr/articles/10.5802/jep.187/
DO  - 10.5802/jep.187
LA  - en
ID  - JEP_2022__9__463_0
ER  - 
%0 Journal Article
%A Coulon, Rémi
%A Steenbock, Markus
%T Product set growth in Burnside groups
%J Journal de l’École polytechnique — Mathématiques
%D 2022
%P 463-504
%V 9
%I École polytechnique
%U http://geodesic.mathdoc.fr/articles/10.5802/jep.187/
%R 10.5802/jep.187
%G en
%F JEP_2022__9__463_0
Coulon, Rémi; Steenbock, Markus. Product set growth in Burnside groups. Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 463-504. doi : 10.5802/jep.187. http://geodesic.mathdoc.fr/articles/10.5802/jep.187/

[Adi79] Adian, S. I. The Burnside problem and identities in groups, Ergeb. Math. Grenzgeb. (3), 95, Springer-Verlag, Berlin-New York, 1979 | MR | DOI

[AL06] Arzhantseva, G. N.; Lysenok, Igor G. A lower bound on the growth of word hyperbolic groups, J. London Math. Soc. (2), Volume 73 (2006) no. 1, pp. 109-125 | Zbl | MR | DOI

[Ata09] Atabekyan, V. S. Uniform nonamenability of subgroups of free Burnside groups of odd period, Mat. Zametki, Volume 85 (2009) no. 4, pp. 516-523 | Zbl | MR | DOI

[BF21] Breuillard, Emmanuel; Fujiwara, Koji On the joint spectral radius for isometries of non-positively curved spaces and uniform growth, Ann. Inst. Fourier (Grenoble), Volume 71 (2021) no. 1, pp. 317-391 | Zbl | MR | DOI

[BG08] Bourgain, Jean; Gamburd, Alex On the spectral gap for finitely-generated subgroups of SU(2), Invent. Math., Volume 171 (2008) no. 1, pp. 83-121 | Zbl | MR | DOI

[BG12] Bourgain, Jean; Gamburd, Alex A spectral gap theorem in SU(d), J. Eur. Math. Soc. (JEMS), Volume 14 (2012) no. 5, pp. 1455-1511 | Zbl | MR | DOI

[BGT12] Breuillard, Emmanuel; Green, Ben; Tao, Terence The structure of approximate groups, Publ. Math. Inst. Hautes Études Sci., Volume 116 (2012), pp. 115-221 | Zbl | MR | DOI

[BH99] Bridson, Martin R.; Haefliger, André Metric spaces of non-positive curvature, Grundlehren Math. Wiss., 319, Springer-Verlag, Berlin, 1999 | DOI

[Bow08] Bowditch, Brian H. Tight geodesics in the curve complex, Invent. Math., Volume 171 (2008) no. 2, pp. 281-300 | Zbl | MR | DOI

[But13] Button, J. O. Explicit Helfgott type growth in free products and in limit groups, J. Algebra, Volume 389 (2013), pp. 61-77 | Zbl | MR | DOI

[CDP90] Coornaert, M.; Delzant, T.; Papadopoulos, A. Géométrie et théorie des groupes. Les groupes hyperboliques de Gromov, Lect. Notes in Math., 1441, Springer-Verlag, Berlin, 1990 | DOI

[Cha08] Chang, Mei-Chu Product theorems in SL 2 and SL 3 , J. Inst. Math. Jussieu, Volume 7 (2008) no. 1, pp. 1-25 | MR | DOI

[Cou13] Coulon, Rémi Growth of periodic quotients of hyperbolic groups, Algebraic Geom. Topol., Volume 13 (2013) no. 6, pp. 3111-3133 | Zbl | MR | DOI

[Cou14] Coulon, Rémi On the geometry of Burnside quotients of torsion free hyperbolic groups, Internat. J. Algebra Comput., Volume 24 (2014) no. 3, pp. 251-345 | Zbl | MR | DOI

[Cou16] Coulon, Rémi Partial periodic quotients of groups acting on a hyperbolic space, Ann. Inst. Fourier (Grenoble), Volume 66 (2016) no. 5, pp. 1773-1857 | Zbl | MR | DOI

[Cou18a] Coulon, Rémi Detecting trivial elements of periodic quotient of hyperbolic groups, Bull. Soc. math. France, Volume 146 (2018) no. 4, pp. 745-806 | Zbl | MR | DOI

[Cou18b] Coulon, Rémi Infinite periodic groups of even exponents, 2018 | arXiv

[DG08] Delzant, Thomas; Gromov, Misha Courbure mésoscopique et théorie de la toute petite simplification, J. Topology, Volume 1 (2008) no. 4, pp. 804-836 | Zbl | DOI

[DGO17] Dahmani, F.; Guirardel, V.; Osin, D. Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces, Mem. Amer. Math. Soc., 245, no. 1156, American Mathematical Society, Providence, RI, 2017

[DS20] Delzant, Thomas; Steenbock, Markus Product set growth in groups and hyperbolic geometry, J. Topology, Volume 13 (2020) no. 3, pp. 1183-1215 | Zbl | MR | DOI

[FS20] Fujiwara, Koji; Sela, Zlil The rates of growth in a hyperbolic group, 2020 | arXiv

[GdlH90] Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Math., 83 (1990) | DOI

[Gro87] Gromov, M. Hyperbolic groups, Essays in group theory (Math. Sci. Res. Inst. Publ.), Volume 8, Springer, New York, 1987, pp. 75-263 | MR | Zbl | DOI

[Hel08] Helfgott, H. A. Growth and generation in SL 2 (ℤ/pℤ), Ann. of Math. (2), Volume 167 (2008) no. 2, pp. 601-623 | MR | DOI

[IO96] Ivanov, Sergei V.; Ol’shanskiĭ, Alexander Yu Hyperbolic groups and their quotients of bounded exponents, Trans. Amer. Math. Soc., Volume 348 (1996) no. 6, pp. 2091-2138 | MR | DOI

[Iva94] Ivanov, Sergei V. The free Burnside groups of sufficiently large exponents, Internat. J. Algebra Comput., Volume 4 (1994) no. 1-2, pp. 1-308 | Zbl | MR | DOI

[Ker21] Kerr, Alice Product set growth in mapping class groups, 2021 | arXiv

[Kou98] Koubi, Malik Croissance uniforme dans les groupes hyperboliques, Ann. Inst. Fourier (Grenoble), Volume 48 (1998) no. 5, pp. 1441-1453 | Zbl | MR | DOI

[Lys96] Lysenok, Igor G. Infinite Burnside groups of even period, Izv. Akad. Nauk SSSR Ser. Mat., Volume 60 (1996) no. 3, pp. 3-224 | DOI

[Nat96] Nathanson, Melvyn B. Additive number theory. Inverse problems and the geometry of sumsets, Graduate Texts in Math., 165, Springer-Verlag, New York, 1996 | DOI

[Ol’82] Ol’shanskiĭ, Alexander Yu. The Novikov-Adyan theorem, Mat. Sb. (N.S.), Volume 118 (1982) no. 2, p. 203-235, 287 | MR

[Ol’91] Ol’shanskiĭ, Alexander Yu. Periodic quotient groups of hyperbolic groups, Mat. Sb. (N.S.), Volume 182 (1991) no. 4, pp. 543-567

[Osi07] Osin, Denis V. Uniform non-amenability of free Burnside groups, Arch. Math. (Basel), Volume 88 (2007) no. 5, pp. 403-412 | MR | Zbl | DOI

[Raz14] Razborov, Alexander A. A product theorem in free groups, Ann. of Math. (2), Volume 179 (2014) no. 2, pp. 405-429 | Zbl | MR | DOI

[Saf11] Safin, S. R. Powers of subsets of free groups, Mat. Sb. (N.S.), Volume 202 (2011) no. 11, pp. 97-102 | DOI

[Sel97] Sela, Z. Acylindrical accessibility for groups, Invent. Math., Volume 129 (1997) no. 3, pp. 527-565 | Zbl | MR | DOI

[Tao08] Tao, Terence Product set estimates for non-commutative groups, Combinatorica, Volume 28 (2008) no. 5, pp. 547-594 | Zbl | MR | DOI

[Tao10] Tao, Terence Freiman’s theorem for solvable groups, Contrib. Discrete Math., Volume 5 (2010) no. 2, pp. 137-184 | Zbl | MR

[TV06] Tao, Terence; Vu, Van Additive combinatorics, Cambridge Studies in Advanced Math., 105, Cambridge University Press, Cambridge, 2006 | DOI

Cité par Sources :