On the entropies of subshifts of finite type on countable amenable groups

Abstract

Let G,H be two countable amenable groups. We introduce the notion of group charts, which gives us a tool to embed an arbitrary H -subshift into a G -subshift. Using an entropy addition formula derived from this formalism we prove that whenever H is finitely presented and admits a subshift of finite type (SFT) on which H acts freely, then the set of real numbers attained as topological entropies of H -SFTs is contained in the set of topological entropies of G -SFTs modulo an arbitrarily small additive constant for any finitely generated group G which admits a translation-like action of H . In particular, we show that the set of topological entropies of G -SFTs on any such group which has decidable word problem and admits a translation-like action of \mathbb{Z}^2 coincides with the set of non-negative upper semi-computable real numbers. We use this result to give a complete characterization of the entropies of SFTs in several classes of groups.