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

Cinzia Bisi; Jean-Philippe Furter; Stéphane Lamy — 2014

Journal de l’École polytechnique — Mathématiques

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.

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