A geometric simulation theorem on direct products of finitely generated groups

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 × H1 × H2)-subshift of finite type for any choice of infinite and finitely generated groups H1, H2. As a consequence, we obtain that every group of the form G1 ×G2 ×G3 admits a non-empty strongly aperiodic SFT subject to the condition that each Gi 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.