Translation-like Actions and Aperiodic Subshifts on Groups

Emmanuel Jeandel (1)

1 CARTE - Theoretical adverse computations, and safety

It is well known that if $G$ admits a f.g. subgroup $H$ with a weakly aperiodic 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.

Domaines: Mathématique discrète [cs.DM]; Théorie des groupes [math.GR]; Systèmes dynamiques [math.DS]

Emmanuel Jeandel. Translation-like Actions and Aperiodic Subshifts on Groups. 2015. ⟨hal-01187069⟩