An embedding theorem for subshifts over amenable groups with the comparison property

Abstract: We obtain the following embedding theorem for symbolic dynamical systems. Let $G$ be a countable amenable group with the comparison property. Let $X$ be a strongly aperiodic subshift over $G$. Let $Y$ be a strongly irreducible shift of finite type over $G$ which has no global period, meaning that the shift action is faithful on $Y$. If the topological entropy of $X$ is strictly less than that of $Y$, and $Y$ contains at least one factor of $X$, then $X$ embeds into $Y$. This result partially extends the classical result of Krieger when $G = \mathbb{Z}$ and the results of Lightwood when $G = \mathbb{Z}^d$ for $d \geq 2$. The proof relies on recent developments in the theory of tilings and quasi-tilings of amenable groups.