Normal amenable subgroups of the automorphism group of sofic shifts

TL;DR

This paper investigates the structure of automorphism groups of sofic shifts, proving that normal amenable subgroups are contained within the shift-generated subgroup in one dimension, but providing a counterexample in higher dimensions.

Contribution

It generalizes a known result about automorphism groups of one-dimensional sofic shifts and demonstrates the limitations of this generalization in higher dimensions.

Findings

Normal amenable subgroups are contained in the shift subgroup for transitive one-dimensional sofic shifts.

Counterexample in two dimensions shows the result does not extend to higher dimensions.

Automorphism group of a 2D mixing shift of finite type can be amenable and not generated by shifts.

Abstract

Let $(X, \sigma)$ be a transitive sofic shift and let $\rm{Aut}(X)$ denote its automorphism group. We generalize a result of Frisch, Schlank, and Tamuz to show that any normal amenable subgroup of $\rm{Aut}(X)$ must be contained in the subgroup generated by the shift. We also show that the result does not extend to higher dimensions by giving an example of a two-dimensional mixing shift of finite type whose automorphism group is amenable and not generated by the shift maps.