The tame automorphism group of an affine quadric threefold acting on a square complex

Cinzia Bisi; Jean-Philippe Furter; Stéphane Lamy

doi:10.5802/jep.8

Cinzia Bisi ¹; Jean-Philippe Furter ²; Stéphane Lamy ³

¹ Dipartimento Matematica ed Informatica, Universitá di Ferrara Via Machiavelli n.35, 44121 Ferrara, Italy
² Laboratoire MIA, Université de La Rochelle Avenue Michel Crépeau, 17000 La Rochelle, France
³ Institut de Mathématiques de Toulouse, Université Paul Sabatier 118 route de Narbonne, 31062 Toulouse Cedex 9, France

Journal de l’École polytechnique — Mathématiques, Volume 1 (2014), pp. 161-223.

Abstract
Résumé

We study the group $Tame ({SL}_{2})$ of tame automorphisms of a smooth affine $3$ -dimensional quadric, which we can view as the underlying variety of ${SL}_{2} (ℂ)$ . We construct a square complex on which the group admits a natural cocompact action, and we prove that the complex is $CAT (0)$ and hyperbolic. We propose two applications of this construction: We show that any finite subgroup in $Tame ({SL}_{2})$ is linearizable, and that $Tame ({SL}_{2})$ satisfies the Tits alternative.

Nous étudions le groupe $Tame ({SL}_{2})$ des automorphismes modérés d’une quadrique affine lisse de dimension $3$ , que l’on peut choisir comme étant la variété sous-jacente à ${SL}_{2} (ℂ)$ . Nous construisons un complexe carré sur lequel ce groupe agit naturellement de façon cocompacte, et nous montrons que ce complexe est $CAT (0)$ et hyperbolique. Nous proposons ensuite deux applications de cette construction : nous montrons que tout sous-groupe fini de $Tame ({SL}_{2})$ est linéarisable, et que $Tame ({SL}_{2})$ satisfait l’alternative de Tits.

Received: 2014-01-09
Accepted: 2014-06-08
Published online: 2014-08-26

MR Zbl

DOI: 10.5802/jep.8

Classification: 14J50, 14R20, 20F65
Keywords: Automorphism group, affine quadric, cube complex, Tits alternative
Mot clés : Groupe d’automorphismes, quadrique affine, complexe cubique, alternative de Tits

Author's affiliations:

Cinzia Bisi ¹; Jean-Philippe Furter ²; Stéphane Lamy ³

¹ Dipartimento Matematica ed Informatica, Universitá di Ferrara Via Machiavelli n.35, 44121 Ferrara, Italy
² Laboratoire MIA, Université de La Rochelle Avenue Michel Crépeau, 17000 La Rochelle, France
³ Institut de Mathématiques de Toulouse, Université Paul Sabatier 118 route de Narbonne, 31062 Toulouse Cedex 9, France

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {The tame automorphism group of an affine quadric threefold acting on a square complex},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {161--223},
     publisher = {\'Ecole polytechnique},
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Cinzia Bisi; Jean-Philippe Furter; Stéphane Lamy. The tame automorphism group of an affine quadric threefold acting on a square complex. Journal de l’École polytechnique — Mathématiques, Volume 1 (2014), pp. 161-223. doi : 10.5802/jep.8. https://jep.centre-mersenne.org/articles/10.5802/jep.8/

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