A note on Automorphisms of the Affine Cremona Group

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 $\theta :\mathcal{G}\to \mathcal{G}$ 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).

5 pages, typos corrected, minor changes, improved exposition

References in corpus (2)

• Wildness of polynomial automorphisms in three variables
• On The Zariski Topology Of Automorphism Groups Of Affine Spaces And Algebras

Cited by in corpus (1)

• Automorphisms of ${\mathbb{C}}^3$ Commuting with a ${\mathbb{C}}^{+}$-Action