Acylindrical hyperbolicity of the three-dimensional tame automorphism group

Lamy, Stéphane; Przytycki, Piotr

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 1, pp. 367-392.

Nous montrons que le groupe $STame\left({𝐤}^3\right)$ 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 $Tame\left({𝐤}^3\right)$ est acylindriquement hyperbolique en exhibant des éléments loxodromiques satisfaisant la propriété WPD.

We prove that the group $STame\left({𝐤}^3\right)$ 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 $Tame\left({𝐤}^3\right)$ is acylindrically hyperbolic by finding explicit loxodromic weakly proper discontinuous elements.

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 SL${}_2\left(𝐂\right)$ , 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