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

### Sebastián Barbieri

Universidad de Santiago de Chile, Chile

## 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 $Z_{2}$ 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.

## Cite this article

Sebastián Barbieri, On the entropies of subshifts of finite type on countable amenable groups. Groups Geom. Dyn. 15 (2021), no. 2, pp. 607–638

DOI 10.4171/GGD/608