Normal subgroup generated by a plane polynomial automorphism

TL;DR

This paper investigates the structure of normal subgroups generated by elements in the group of complex plane polynomial automorphisms with Jacobian 1, revealing conditions under which these subgroups are either the entire group or proper subsets.

Contribution

It provides new insights into the generation of normal subgroups in the automorphism group, especially relating subgroup size to the length of the generating element.

Findings

If the element has length ≤ 8, the generated subgroup equals the entire group.

For generic elements of even length ≥ 14, the generated subgroup is proper.

The subgroup structure depends on the length and genericity of the automorphism.

Abstract

We study the normal subgroup <f> generated by a non trivial element f in the group G of complex plane polynomial automorphisms having Jacobian determinant 1. On one hand if f has length at most 8 relatively to the classical amalgamated product structure of G, we prove that <f> = G. On the other hand if f is a sufficiently generic element of even length at least 14, we prove that <f> is a proper subgroup of G.