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

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