Acylindrical hyperbolicity of the three-dimensional tame automorphism group
[Hyperbolicité acylindrique du groupe des automorphismes modérés en dimension 3]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 1, pp. 367-392.

Nous montrons que le groupe STame(𝐤3) des automorphismes modérés unimodulaires de l'espace affine de dimension 3 n'est pas simple, sur tout corps de base de caractéristique zéro. Notre preuve repose sur l'étude géométrique d'un complexe simplicial 𝒞 simplement connexe et de dimension 2, sur lequel le groupe des automorphismes modérés agit naturellement. Nous montrons que 𝒞 est contractible et hyperbolique au sens de Gromov, puis nous prouvons que Tame(𝐤3) est acylindriquement hyperbolique en exhibant des éléments loxodromiques satisfaisant la propriété WPD.

We prove that the group STame(𝐤3) of special tame automorphisms of the affine 3-space is not simple, over any base field of characteristic zero. Our proof is based on the study of the geometry of a 2-dimensional simply-connected simplicial complex 𝒞 on which the tame automorphism group acts naturally. We prove that 𝒞 is contractible and Gromov-hyperbolic, and we prove that Tame(𝐤3) is acylindrically hyperbolic by finding explicit loxodromic weakly proper discontinuous elements.

DOI : 10.24033/asens.2390
Classification : 14R10, 20F65, 57M20
Keywords: Tame automorphisms, acylindrical hyperbolicity, triangle complex
Mot clés : Automorphismes mod/'er/'es, hyperbolicit/'e acylindrique, complexe de triangles 
@article{ASENS_2019__52_1_369_0,
     author = {Lamy, St\'ephane and Przytycki, Piotr},
     title = {Acylindrical hyperbolicity  of the three-dimensional  tame automorphism group},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {367--392},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 52},
     number = {1},
     year = {2019},
     doi = {10.24033/asens.2390},
     mrnumber = {3948112},
     zbl = {1442.14188},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/asens.2390/}
}
TY  - JOUR
AU  - Lamy, Stéphane
AU  - Przytycki, Piotr
TI  - Acylindrical hyperbolicity  of the three-dimensional  tame automorphism group
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2019
SP  - 367
EP  - 392
VL  - 52
IS  - 1
PB  - Société Mathématique de France. Tous droits réservés
UR  - https://www.numdam.org/articles/10.24033/asens.2390/
DO  - 10.24033/asens.2390
LA  - en
ID  - ASENS_2019__52_1_369_0
ER  - 
%0 Journal Article
%A Lamy, Stéphane
%A Przytycki, Piotr
%T Acylindrical hyperbolicity  of the three-dimensional  tame automorphism group
%J Annales scientifiques de l'École Normale Supérieure
%D 2019
%P 367-392
%V 52
%N 1
%I Société Mathématique de France. Tous droits réservés
%U https://www.numdam.org/articles/10.24033/asens.2390/
%R 10.24033/asens.2390
%G en
%F ASENS_2019__52_1_369_0
Lamy, Stéphane; Przytycki, Piotr. Acylindrical hyperbolicity  of the three-dimensional  tame automorphism group. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 1, pp. 367-392. doi : 10.24033/asens.2390. https://www.numdam.org/articles/10.24033/asens.2390/

Bestvina, M.; Fujiwara, K. Bounded cohomology of subgroups of mapping class groups, Geom. Topol., Volume 6 (2002), pp. 69-89 | DOI | MR | Zbl

Bisi, C.; Furter, J.-P.; Lamy, S. The tame automorphism group of an affine quadric threefold acting on a square complex, J. Éc. polytech. Math., Volume 1 (2014), pp. 161-223 | DOI | MR | Zbl

Bridson, M. R.; Haefliger, A., Grundl. math. Wiss., 319, Springer, 1999, 643 pages | MR | Zbl

Bowditch, B. H. Tight geodesics in the curve complex, Invent. math., Volume 171 (2008), pp. 281-300 | DOI | MR | Zbl

Cantat, S. Sur les groupes de transformations birationnelles des surfaces, Ann. of Math., Volume 174 (2011), pp. 299-340 | DOI | MR | Zbl

Cantat, S., European Congress of Mathematics, Eur. Math. Soc., 2013, pp. 211-225 | MR | Zbl

Cantat, S.; Lamy, S. Normal subgroups in the Cremona group, Acta Math., Volume 210 (2013), pp. 31-94 | DOI | MR | Zbl

Danilov, V. I. Non-simplicity of the group of unimodular automorphisms of an affine plane, Mat. Zametki, Volume 15 (1974), pp. 289-293 | MR

Dahmani, F.; Guirardel, V.; Osin, D. Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces, Mem. Amer. Math. Soc., Volume 245 (2017) | MR

Furter, J.-P.; Lamy, S. Normal subgroup generated by a plane polynomial automorphism, Transform. Groups, Volume 15 (2010), pp. 577-610 | DOI | MR | Zbl

Januszkiewicz, T.; Świątkowski, J. Simplicial nonpositive curvature, Publ. Math. Inst. Hautes Études Sci., Volume 104, pp. 1-85 | DOI | Numdam | MR | Zbl

Kuroda, S. Shestakov-Umirbaev reductions and Nagata's conjecture on a polynomial automorphism, Tohoku Math. J., Volume 62 (2010), pp. 75-115 | DOI | MR | Zbl

Lamy, S. Combinatorics of the tame automorphism group ( arXiv:1505.0549, to appear in Ann. Fac. Sci. Toulouse ) | MR

Lonjou, A. Non simplicité du groupe de Cremona sur tout corps, Ann. Inst. Fourier, Volume 66 (2016), pp. 2021-2046 | DOI | MR | Zbl

Lyndon, R. C.; Schupp, P. E., Ergebn. Math. Grenzg., 89, Springer, 1977, 339 pages | MR | Zbl

Martin, A. On the acylindrical hyperbolicity of the tame automorphism group of SL2(𝐂) , Bull. London Math. Soc., Volume 49 (2017), pp. 881-894 | DOI | MR | Zbl

Minasyan, A.; Osin, D. Acylindrical hyperbolicity of groups acting on trees, Math. Ann., Volume 362 (2015), pp. 1055-1105 | DOI | MR | Zbl

Osin, D. Acylindrically hyperbolic groups, Trans. Amer. Math. Soc., Volume 368 (2016), pp. 851-888 | DOI | MR | Zbl

Serre, J.-P., Springer, 1980 | MR | Zbl

Shestakov, I. P.; Umirbaev, U. U. The tame and the wild automorphisms of polynomial rings in three variables, J. Amer. Math. Soc., Volume 17 (2004), pp. 197-227 | DOI | MR | Zbl

Wright, D. The generalized amalgamated product structure of the tame automorphism group in dimension three, Transform. Groups, Volume 20 (2015), pp. 291-304 | DOI | MR | Zbl

Cité par Sources :