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We describe the structure of the group of algebraic automorphisms of the following surfaces 1) P1,k x P1,k minus a diagonal; 2) P1,k x P1,k minus a fiber. The motivation is to get a new proof of two theorems proven respectively by L. Makar-Limanov and H. Nagao. We also discuss the structure of the semi-group of polynomial proper maps from C2 to C2.

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.