Distortion and the automorphism group of a shift

TL;DR

This paper investigates the structure of automorphism groups of zero entropy shifts, demonstrating that certain complex groups, including Baumslag-Solitar groups, cannot embed into these automorphism groups due to distortion properties.

Contribution

It provides the first examples of countable groups that cannot embed into automorphism groups of zero entropy shifts, highlighting the role of distortion in such embeddings.

Findings

Baumslag-Solitar groups cannot embed into automorphism groups of zero entropy shifts.

Distortion in nilpotent groups obstructs embedding into low complexity shifts.

Automorphism groups of zero entropy shifts are more restricted than previously thought.

Abstract

The set of automorphisms of a one-dimensional \shift $(X, \sigma)$ forms a countable, but often very complicated, group. For zero entropy shifts, it has recently been shown that the automorphism group is more tame. We provide the first examples of countable groups that cannot embed into the automorphism group of any zero entropy \shiftno. In particular, we show that the Baumslag-Solitar groups ${\rm BS}(1,n)$ and all other groups that contain exponentially distorted elements cannot embed into ${\rm Aut}(X)$ when $h_{{\rm top}}(X)=0$. We further show that distortion in nilpotent groups gives a nontrivial obstruction to embedding such a group in any low complexity shift.