Product set growth in Burnside groups

Rémi Coulon; Markus Steenbock

doi:10.5802/jep.187

Product set growth in Burnside groups
[Croissance des ensembles produit dans les groupes de Burnside]

Rémi Coulon ¹ ; Markus Steenbock ²

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.

Résumé
Abstract

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.

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.

Reçu le : 2021-02-26
Accepté le : 2022-02-15
Publié le : 2022-02-28

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
Mot clés : Ensemble produit, croissance, groupes hyperboliques, actions cylindriques, théorie de la petite simplification, groupes périodiques infinis, problème de Burnside

Affiliations des auteurs :

Rémi Coulon ¹ ; Markus Steenbock 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 = {R\'emi Coulon and Markus Steenbock},
     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 = {https://jep.centre-mersenne.org/articles/10.5802/jep.187/}
}

TY  - JOUR
AU  - Rémi Coulon
AU  - Markus Steenbock
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  - https://jep.centre-mersenne.org/articles/10.5802/jep.187/
DO  - 10.5802/jep.187
LA  - en
ID  - JEP_2022__9__463_0
ER  -

%0 Journal Article
%A Rémi Coulon
%A Markus Steenbock
%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 https://jep.centre-mersenne.org/articles/10.5802/jep.187/
%R 10.5802/jep.187
%G en
%F JEP_2022__9__463_0

Rémi Coulon; Markus Steenbock. Product set growth in Burnside groups. Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 463-504. doi : 10.5802/jep.187. https://jep.centre-mersenne.org/articles/10.5802/jep.187/

Bibliographie
Cité par

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

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

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

[BF21] E. Breuillard & K. Fujiwara - “On the joint spectral radius for isometries of non-positively curved spaces and uniform growth”, Ann. Inst. Fourier (Grenoble) 71 (2021) no. 1, p. 317-391 | DOI | MR | Zbl

[BG08] J. Bourgain & A. Gamburd - “On the spectral gap for finitely-generated subgroups of $SU (2)$ ”, Invent. Math. 171 (2008) no. 1, p. 83-121 | DOI | MR | Zbl

[BG12] J. Bourgain & A. Gamburd - “A spectral gap theorem in $SU (d)$ ”, J. Eur. Math. Soc. (JEMS) 14 (2012) no. 5, p. 1455-1511 | DOI | MR | Zbl

[BGT12] E. Breuillard, B. Green & T. Tao - “The structure of approximate groups”, Publ. Math. Inst. Hautes Études Sci. 116 (2012), p. 115-221 | DOI | MR | Zbl

[BH99] M. R. Bridson & A. Haefliger - Metric spaces of non-positive curvature, Grundlehren Math. Wiss., vol. 319, Springer-Verlag, Berlin, 1999 | DOI

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

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

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

[Cha08] M.-C. Chang - “Product theorems in ${SL}_{2}$ and ${SL}_{3}$ ”, J. Inst. Math. Jussieu 7 (2008) no. 1, p. 1-25 | DOI | MR

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

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

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

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

[Cou18b] R. Coulon - “Infinite periodic groups of even exponents”, 2018 | arXiv

[DG08] T. Delzant & M. Gromov - “Courbure mésoscopique et théorie de la toute petite simplification”, J. Topology 1 (2008) no. 4, p. 804-836 | DOI | Zbl

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

[DS20] T. Delzant & M. Steenbock - “Product set growth in groups and hyperbolic geometry”, J. Topology 13 (2020) no. 3, p. 1183-1215 | DOI | MR | Zbl

[FS20] K. Fujiwara & Z. Sela - “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] M. Gromov - “Hyperbolic groups”, in Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, p. 75-263 | DOI | MR | Zbl

[Hel08] H. A. Helfgott - “Growth and generation in ${SL}_{2} (ℤ / p ℤ)$ ”, Ann. of Math. (2) 167 (2008) no. 2, p. 601-623 | DOI | MR

[IO96] S. V. Ivanov & A. Y. Ol’shanskiĭ - “Hyperbolic groups and their quotients of bounded exponents”, Trans. Amer. Math. Soc. 348 (1996) no. 6, p. 2091-2138 | DOI | MR

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

[Ker21] A. Kerr - “Product set growth in mapping class groups”, 2021 | arXiv

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

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

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

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

[Ol’91] A. Y. Ol’shanskiĭ - “Periodic quotient groups of hyperbolic groups”, Mat. Sb. (N.S.) 182 (1991) no. 4, p. 543-567

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

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

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

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

[Tao08] T. Tao - “Product set estimates for non-commutative groups”, Combinatorica 28 (2008) no. 5, p. 547-594 | DOI | MR | Zbl

[Tao10] T. Tao - “Freiman’s theorem for solvable groups”, Contrib. Discrete Math. 5 (2010) no. 2, p. 137-184 | MR | Zbl

[TV06] T. Tao & V. Vu - Additive combinatorics, Cambridge Studies in Advanced Math., vol. 105, Cambridge University Press, Cambridge, 2006 | DOI

Cité par Sources :