The automorphism group of a shift of slow growth is amenable

TL;DR

This paper proves that for subshifts with slow growth in complexity, the automorphism group is amenable and cannot contain free semigroups, contrasting with recent examples of quadratic complexity subshifts.

Contribution

It establishes new conditions under which the automorphism group of a subshift is amenable or lacks free semigroups, based on the complexity growth rate.

Findings

If $P_X(n)=o(n^2/ ext{log}^2 n)$, then ${ m Aut}(X)$ is amenable.

If $P_X(n)=o(n^2)$, then ${ m Aut}(X)$ cannot contain a nonabelian free semigroup.

Contrasts with recent examples of quadratic complexity subshifts containing free semigroups.

Abstract

Suppose $(X,\sigma)$ is a subshift, $P_X(n)$ is the word complexity function of $X$, and ${\rm Aut}(X)$ is the group of automorphisms of $X$. We show that if $P_X(n)=o(n^2/\log^2 n)$, then ${\rm Aut}(X)$ is amenable (as a countable, discrete group). We further show that if $P_X(n)=o(n^2)$, then ${\rm Aut}(X)$ can never contain a nonabelian free semigroup (and, in particular, can never contain a nonabelian free subgroup). This is in contrast to recent examples, due to Salo and Schraudner, of subshifts with quadratic complexity that do contain such a semigroup.