[Hyperbolicité acylindrique du groupe des automorphismes modérés en dimension 3]
Nous montrons que le groupe 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 simplement connexe et de dimension 2, sur lequel le groupe des automorphismes modérés agit naturellement. Nous montrons que est contractible et hyperbolique au sens de Gromov, puis nous prouvons que est acylindriquement hyperbolique en exhibant des éléments loxodromiques satisfaisant la propriété WPD.
We prove that the group 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 on which the tame automorphism group acts naturally. We prove that is contractible and Gromov-hyperbolic, and we prove that is acylindrically hyperbolic by finding explicit loxodromic weakly proper discontinuous elements.
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 = {http://archive.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 - http://archive.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 http://archive.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. http://archive.numdam.org/articles/10.24033/asens.2390/
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