A note on automorphisms of the affine Cremona group

Let $\mathcal{G}$ be an ind-group and let $\mathcal{U \subseteq G}$ be a unipotent ind-subgroup. We prove that an abstract group automorphism $\mathcal{\theta \colon G \to G}$ maps $\mathcal{U}$ isomorphically onto a unipotent ind-subgroup of $\mathcal{G}$, provided that $\mathcal{\theta}$ fixes a closed torus $\mathcal{T \subseteq G}$, which normalizes $\mathcal{U}$ and the action of $\mathcal{T}$ on $\mathcal{U}$ by conjugation fixes only the neutral element. As an application we generalize a result by Hanspeter Kraft and the author as follows: If an abstract group automorphism of the affine Cremona group $\mathcal{G_3}$ in dimension $\mathcal{3}$ fixes the subgroup of tame automorphisms $\mathcal{TG_3}$, then it also fixes a whole family of non-tame automorphisms (including the Nagata automorphism).

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Published 13 June 2014