Translation-like Actions and Aperiodic Subshifts on Groups

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 G1 and G2 are f.g. infinite groups, then G1 × G2 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. A subshift of finite type over a group G corresponds to a description of colorings of the vertices of its Cayley graph subject to local constraints. Even for harmless groups like Z, it is possible to build [4] easily [12] subshifts with no periodic points. For this group, the domino problem, which consists in deciding if a subshift of finite type is empty, is even undecidable [4]. In this article, we are interested in which groups enjoy similar properties: In which groups can we build aperiodic subshifts of finite type, and which groups have an undecidable word problem. There are various definitions of “aperiodicity” on a group, and here we study weakly aperiodic subshifts: no coloring has a finite orbit. Apart from the example of Z, aperiodic subshifts have been built on BaumslagSolitar Groups [1], on the free group [17] and on every group of nonlinear polynomial growth [2, 6]. In all these examples except the free group, this actually gives groups with an undecidable domino problem. It is easy to see that if a f.g. group G has an aperiodic SFT, then every group that contains G also has an aperiodic SFT. In this statement, “contains” means subgroup containment. The goal of this article is to prove that this is