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

Bisi, Cinzia; Furter, Jean-Philippe; Lamy, Stéphane

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

Nous étudions le groupe $\mathrm{Tame}\left({\mathrm{SL}}_2\right)$ des automorphismes modérés d’une quadrique affine lisse de dimension $3$, que l’on peut choisir comme étant la variété sous-jacente à ${\mathrm{SL}}_2\left(ℂ\right)$. Nous construisons un complexe carré sur lequel ce groupe agit naturellement de façon cocompacte, et nous montrons que ce complexe est $\mathrm{CAT}\left(0\right)$ et hyperbolique. Nous proposons ensuite deux applications de cette construction : nous montrons que tout sous-groupe fini de $\mathrm{Tame}\left({\mathrm{SL}}_2\right)$ est linéarisable, et que $\mathrm{Tame}\left({\mathrm{SL}}_2\right)$ satisfait l’alternative de Tits.

We study the group $\mathrm{Tame}\left({\mathrm{SL}}_2\right)$ of tame automorphisms of a smooth affine $3$-dimensional quadric, which we can view as the underlying variety of ${\mathrm{SL}}_2\left(ℂ\right)$. We construct a square complex on which the group admits a natural cocompact action, and we prove that the complex is $\mathrm{CAT}\left(0\right)$ and hyperbolic. We propose two applications of this construction: We show that any finite subgroup in $\mathrm{Tame}\left({\mathrm{SL}}_2\right)$ is linearizable, and that $\mathrm{Tame}\left({\mathrm{SL}}_2\right)$ satisfies the Tits alternative.

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

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