On Automorphisms of the Affine Cremona Group

Abstract

We show that every automorphism of the group ${\mathcal{G}}_n:=\text{Aut}({\mathbb{A}}^n)$ of polynomial automorphisms of complex affine $n$-space ${\mathbb{A}}^n={\mathbb{C}}^n$ is inner up to field automorphisms when restricted to the subgroup $T{\mathcal{G}}_n$ of tame automorphisms. This generalizes a result of \textsc{Julie Deserti} who proved this in dimension $n=2$ where all automorphisms are tame: $T{\mathcal{G}}_2={\mathcal{G}}_2$.

8 pages, typos corrected, minor changes, improved exposition

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