Distortion and the automorphism group of a shift

Van Cyr,Bryna Kra,Samuel Petite,John Franks

doi:10.3934/jmd.2018015

Abstract

The set of automorphisms of a one-dimensional subshift \begin{document} $(X, σ)$ \end{document} 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 subshift. In particular, we show that the Baumslag-Solitar groups \begin{document} ${\rm BS}(1,n)$ \end{document} and all other groups that contain exponentially distorted elements cannot embed into \begin{document} ${\rm Aut}(X)$ \end{document} when \begin{document} $h_{{\rm top}}(X) = 0$ \end{document} . We further show that distortion in nilpotent groups gives a nontrivial obstruction to embedding such a group in any low complexity shift.

Highlights

If Σ is a finite alphabet and X ⊂ ΣZ is a closed set that is invariant under the left shift σ : ΣZ → ΣZ, (X, σ) is called a subshift
Theorems of this nature typically take the following form: suppose (X, σ) is a subshift with some dynamical assumption and suppose that the complexity function of (X, σ) grows more slowly than some explicitly chosen subexponential rate, Aut(X ) has some particular algebraic property. Without these growth rate and dynamical assumptions, little is known about the algebraic structure of Aut(X ). It was asked in [7, Question 6.1] whether every countable group arises as the automorphism group of some minimal, zero entropy shift
For very low complexity systems, we improve on this result, showing that for any shift whose complexity function is o(n((d+1)(d+2)/2)+2), any finitely generated, torsion free subgroup of the automorphism group is virtually d -step nilpotent

Summary

INTRODUCTION

If Σ is a finite alphabet and X ⊂ ΣZ is a closed set that is invariant under the left shift σ : ΣZ → ΣZ, (X , σ) is called a subshift. For shifts with low complexity (see Section 2 for precise definitions), there are numerous restrictions that arise (see [5, 6, 7]) Theorems of this nature typically take the following form: suppose (X , σ) is a subshift with some dynamical assumption (such as minimality or transitivity) and suppose that the complexity function of (X , σ) grows more slowly than some explicitly chosen subexponential rate, Aut(X ) has some particular algebraic property. For very low complexity systems, we improve on this result, showing that for any shift whose complexity function is o(n((d+1)(d+2)/2)+2), any finitely generated, torsion free subgroup of the automorphism group is virtually d -step nilpotent (the precise statement is in Theorem 4.10). We conclude with several open questions, primarily on what sorts of restrictions can be placed on the automorphism group of a shift

BACKGROUND

SUBGROUPS OF THE AUTOMORPHISM GROUP

TORSION FREE NILPOTENT GROUPS

OPEN QUESTIONS