Topological simplicity of the group of automorphisms of the affine plane

We prove that the group ${\mathrm{S}\mathrm{A}\mathrm{u}\mathrm{t}}_{\mathrm{k}}({\mathbb{A}}^2)$ is simple as an algebraic group of infinite dimension, over any infinite field $\mathrm{k}$, by proving that any closed normal subgroup is either trivial or the whole group. In higher dimension, we show that closed normal subgroups contain all tame automorphisms. The case of finite fields, very different, is also discussed.