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

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 that 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 = Z and the results of Lightwood when G = Zᵈ for d ≥ 2 . The proof relies on recent developments in the theory of tilings and quasi-tilings of amenable groups.