The affine automorphism group of A^3 is not a maximal subgroup of the tame automorphism group

TL;DR

This paper constructs explicit subgroups of the tame automorphism group of affine three-space, demonstrating that the affine automorphism group is not maximal within the tame automorphism group, and describes their algebraic structure.

Contribution

It provides explicit examples of proper subgroups generated by affine and non-affine automorphisms, answering an open question about maximality in characteristic zero.

Findings

Affine automorphism group is not maximal in the tame automorphism group.

Constructed subgroups are amalgamated free products of affine and finite groups.

Subgroups are generated by affine and specific non-affine automorphisms.

Abstract

We construct explicitly a family of proper subgroups of the tame automorphism group of affine three-space (in any characteristic) which are generated by the affine subgroup and a non-affine tame automorphism. One important corollary is the titular result that settles negatively the open question (in characteristic zero) of whether the affine subgroup is a maximal subgroup of the tame automorphism group. We also prove that all groups of this family have the structure of an amalgamated free product of the affine group and a finite group over their intersection.