Topological simplicity of the group of automorphisms of the affine plane

TL;DR

This paper proves the algebraic simplicity of the automorphism group of the affine plane over infinite fields and discusses the structure of normal subgroups in higher dimensions and finite fields.

Contribution

It establishes the simplicity of the automorphism group of the affine plane as an algebraic group over infinite fields and explores the subgroup structure in higher dimensions and finite fields.

Findings

The automorphism group of the affine plane is simple over infinite fields.

Normal subgroups in higher dimensions contain all tame automorphisms.

Finite field case differs significantly from the infinite field case.

Abstract

We prove that the group $\mathrm{SAut}_{\mathrm{k}}(\mathbb{A}^2)$ is simple as an algebraic group of infinite dimension, over any infinite field $\mathrm{k}$, by proving that any closed normal subgroup is either trivial or the whole group. In higher dimension, we show that closed normal subgroups contain all tame automorphisms. The case of finite fields, very different, is also discussed.