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.
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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 
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     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},
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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. http://www.numdam.org/articles/10.24033/asens.2390/

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