The automorphism group of a shift of linear growth: beyond transitivity

Abstract

For a finite alphabet $\mathcal{A}$ and shift $X\subseteq {\mathcal{A}}^{\mathbb{Z}}$ whose factor complexity function grows at most linearly, we study the algebraic properties of the automorphism group $\mathrm{Aut}(X)$. For such systems, we show that every finitely generated subgroup of $\mathrm{Aut}(X)$ is virtually ${\mathbb{Z}}^d$, in contrast to the behavior when the complexity function grows more quickly. With additional dynamical assumptions we show more: if $X$ is transitive, then $\mathrm{Aut}(X)$ is virtually $\mathbb{Z}$; if $X$ has dense aperiodic points, then $\mathrm{Aut}(X)$ is virtually ${\mathbb{Z}}^d$. We also classify all finite groups that arise as the automorphism group of a shift.