Acylindrical hyperbolicity of the three-dimensional tame automorphism group

TL;DR

This paper demonstrates that the group of special tame automorphisms of affine 3-space is not simple by analyzing its action on a hyperbolic simplicial complex, establishing its acylindrical hyperbolicity.

Contribution

It introduces a geometric approach to prove acylindrical hyperbolicity of the tame automorphism group of affine 3-space over characteristic zero fields.

Findings

The simplicial complex C is contractible and Gromov-hyperbolic.

The tame automorphism group acts on C with acylindrical hyperbolicity.

Explicit loxodromic elements are identified within the group.

Abstract

We prove that the group STame($k^3$) of special tame automorphisms of the affine 3-space is not simple, over any base field of characteristic zero. Our proof is based on the study of the geometry of a 2-dimensional simply-connected simplicial complex C on which the tame automorphism group acts naturally. We prove that C is contractible and Gromov-hyperbolic, and we prove that Tame($k^3$) is acylindrically hyperbolic by finding explicit loxodromic weakly proper discontinuous elements.