The affine automorphism group of A³ is not a maximal subgroup of the tame automorphism group

2014/10/03 by Éric Edo, Drew Lewis, Edo, Eric +1
Mathematics · Medicine · #14R10 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models

paper · pdf · doi:10.48550/arxiv.1410.0901

openalex publication_date 2014/10/03 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

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.

Related

• On the closure of the tame automorphism group of affine three-space
• The Amalgamated Product Structure of the Tame Automorphism Group in Dimension Three
• Co-tame polynomial automorphisms
• Closed subgroups of the polynomial automorphism group containing the affine subgroup
• Families of commuting automorphisms, and a characterization of the affine space