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Published online by Cambridge University Press: 02 March 2015
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
$\text{Aut}(X)$. For such systems, we show that every finitely generated subgroup of
$\text{Aut}(X)$ 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
$\text{Aut}(X)$ is virtually
$\mathbb{Z}$; if
$X$ has dense aperiodic points, then
$\text{Aut}(X)$ is virtually
$\mathbb{Z}^{d}$. We also classify all finite groups that arise as the automorphism group of a shift.