In this article, we prove that a $\mathbb{Q}$-homology plane $X$ with two algebraically independent $G_a$-actions is isomorphic to either the affine plane or a quotient of an affine hypersurface $xy=z^m-1$ in the affine $3$-space via a free $\mathbb{Z}/m\mathbb{Z}$-action, where $m$ is the order of a finite group $H_1(X;\mathbb{Z})$.

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.