On Automorphisms of the Affine Cremona Group

TL;DR

This paper proves that all automorphisms of the polynomial automorphism group of complex affine space are essentially inner, up to field automorphisms, when considering the subgroup of tame automorphisms, extending previous results from dimension two.

Contribution

It generalizes Julie Deserti's result by showing automorphisms are inner up to field automorphisms for all dimensions, within the tame automorphism subgroup.

Findings

Automorphisms of the tame subgroup are inner up to field automorphisms.

Extension of Deserti's result from dimension 2 to higher dimensions.

Provides structural insight into the automorphism group of polynomial automorphisms.

Abstract

We show that every automorphism of the group $\mathcal{G}_n:= \textrm{Aut}(\mathbb{A}^n)$ 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: $T \mathcal{G}_2 = \mathcal{G}_2$.