  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,$\mathbb{C}$). 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.
