Normal subgroups generated by a single polynomial automorphism

TL;DR

This paper investigates when a single polynomial automorphism generates the entire normal subgroup related to the special linear group, showing that certain automorphisms in characteristic zero generate the full normal closure in higher dimensions.

Contribution

It extends previous results by proving that nontrivial 4-triangular automorphisms generate the entire normal closure of the special linear group in any dimension over characteristic zero fields.

Findings

Nontrivial 4-triangular automorphisms generate the full normal closure.

Generalization of Furter and Lamy's result to higher dimensions.

Results hold for any dimension n ≥ 2 in characteristic zero.

Abstract

We study criteria for deciding when the normal subgroup generated by a single polynomial automorphism of $\mathbb{A}^n$ is as large as possible, namely equal to the normal closure of the special linear group in the special automorphism group. In particular, we investigate $m$-triangular automorphisms, i.e. those that can be expressed as a product of affine automorphisms and $m$ triangular automorphisms. Over a field of characteristic zero, we show that every nontrivial $4$-triangular special automorphism generates the entire normal closure of the special linear group in the special tame subgroup, for any dimension $n \geq 2$. This generalizes a result of Furter and Lamy in dimension 2.