A note on Automorphisms of the Affine Cremona Group

TL;DR

This paper studies automorphisms of ind-groups, showing they preserve unipotent subgroups under certain conditions, and applies this to the affine Cremona group, demonstrating automorphisms fixing tame automorphisms also fix non-tame ones.

Contribution

It generalizes previous results by proving automorphisms of ind-groups preserve unipotent subgroups when fixing a torus, and applies this to the affine Cremona group to relate tame and non-tame automorphisms.

Findings

Automorphisms map unipotent subgroups isomorphically under certain conditions.

Automorphisms fixing a torus also fix a family of non-tame automorphisms.

Extension of Kraft's result to higher complexity automorphisms.

Abstract

Let $\mathcal{G}$ be an ind-group and let $\mathcal{U} \subseteq \mathcal{G}$ be a unipotent ind-subgroup. We prove that an abstract group automorphism $\theta \colon \mathcal{G} \to \mathcal{G}$ maps $\mathcal{U}$ isomorphically onto a unipotent ind-subgroup of $\mathcal{G}$, provided that $\theta$ fixes a closed torus $T \subseteq \mathcal{G}$, which normalizes $\mathcal{U}$ and the action of $T$ 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 3 fixes the subgroup of tame automorphisms $T{\mathcal G}_3$, then it also fixes a whole family of non-tame automorphisms (including the Nagata automorphism).