Aperiodic Subshifts on Polycyclic Groups

We prove that every polycyclic group of nonlinear growth admits a strongly aperiodic SFT and has an undecidable domino problem. This answers a question of [5] and generalizes the result of [2]. Subshifts of finite type (SFT for short) in a group G are colorings of the elements of G subject to local constraints. They have been studied extensively for the free abelian group Z [14] and on the free abelian group Z, where they correspond to tilings of the discrete plane [13]. One of the main question about SFTs is the existence of strongly aperiodic SFT, that is finding a finite set of local constraints so that the only way to color the group G is to do so aperiodically. Strongly aperiodic SFT are known to exists for the group Z [4] and the author [10] has provided examples on many groups of the form Z×G. A related question is the decidability of the domino problem: Given an SFT X on a group G, decide if X is empty. This question is somewhat related to the existence of weakly aperiodic SFT, i.e. where no coloring is periodic along a subgroup of finite index in G. The undecidability of the domino problem has been established on Z [4], on Baumslag-Solitar groups [1] and on any nilpotent group of nonlinear growth [2]. In this article we introduce a few elementary techniques that show how to exhibit strongly aperiodic SFT on any polycyclic group which is not of linear growth (i.e. not virtually Z), and we also prove that any such group has an undecidable domino problem. The first result answers a question of Carroll and Penland [5], while the second generalizes the result of Ballier and Stein from nilpotent groups to polycyclic groups. A full characterization of groups (or even finitely presented groups) with strongly aperiodic SFT is still for now out of reach.