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

Published in Michigan Mathematics Journal, 2015

Recommended citation:

E. Edo & D. Lewis, The affine automorphism group of $\mathbb{A}^3$ is not a maximal subgroup of the tame automorphism group, Michigan Mathematics Journal, 64(3) (2015), 555-568.

### Abstract

We construct explicitly a family of proper subgroups of the tame automorphism group of affine three-space (in any characteristic) that are generated by the affine subgroup and a nonaffine 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.