Combinatorics of the tame automorphism group

TL;DR

This paper investigates the structure of the tame automorphism group of three-dimensional affine space over characteristic zero fields, introducing a new geometric perspective using a simply connected 2D simplicial complex.

Contribution

It provides a unified, simplified approach to existing results on reductions and relations in Tame(A^3), emphasizing a geometric action on a simplicial complex.

Findings

Reconstruction of previous results on the structure of Tame(A^3)

Introduction of a simply connected 2D simplicial complex for group action

Simplified and unified presentation of the theory of reduction and relations

Abstract

We study the group Tame($\mathbf A^3$) of tame automorphisms of the 3-dimensional affine space, over a field of characteristic zero. We recover, in a unified and (hopefully) simplified way, previous results of Kuroda, Shestakov, Umirbaev and Wright, about the theory of reduction and the relations in Tame($\mathbf A^3$). The novelty in our presentation is the emphasis on a simply connected 2-dimensional simplicial complex on which Tame($\mathbf A^3$) acts by isometries.