A geometric simulation theorem on direct products of finitely generated groups

Sebastián Barbieri

Published 2017-06-02Version 1

We show that every effectively closed action of a finitely generated group $G$ over a Cantor set can be obtained as a topological factor of the $G$-subaction of a $(G \times H_1 \times H_2)$-subshift of finite type for any choice of infinite and finitely generated groups $H_1,H_2$. As a consequence, we obtain that every group of the form $G_1 \times G_2 \times G_3$ admits a non-empty strongly aperiodic SFT subject to the condition that each $G_i$ is finitely generated and has decidable word problem. As a corollary of this last result we prove the existence of a non-empty strongly aperiodic SFT in the Grigorchuk group.