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We study the group Tame(SL$_2$) of tame automorphisms of a smooth affine3-dimensional quadric, which we can view as the underlying variety ofSL(2,$\mathbb{C}$). We construct a square complex on which the group admits anatural cocompact action, and we prove that the complex is CAT(0) andhyperbolic. We propose two applications of this construction: We show that anyfinite subgroup in Tame(SL$_2$) is linearizable, and that Tame(SL$_2$)satisfies the Tits alternative.

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