Translation-like Actions and Aperiodic Subshifts on Groups

Abstract

It is well known that if $G$ admits a f.g. subgroup $H$ with a weaklyaperiodic SFT (resp. an undecidable domino problem), then $G$itself has a weakly aperiodic SFT (resp. an undecidable domino problem).We prove that we can replace the property "$H$ is a subgroup of $G$"by "$H$ acts translation-like on $G$", provided $H$ is finitely presented.In particular:* If $G\_1$ and $G\_2$ are f.g. infinite groups, then $G\_1 \times G\_2$ has a weakly aperiodic SFT (and actually a undecidable domino problem). In particular the Grigorchuk group has an undecidable domino problem. * Every infinite f.g. $p$-group admits a weakly aperiodic SFT.