Aperiodic Subshifts of Finite Type on Groups

In this note we prove the following results: • If a finitely presented group G admits a strongly aperiodic SFT, then G has decidable word problem. • For a large class of group G, Z × G admits a strongly aperiodic SFT. In particular, this is true for the free group with 2 generators, Thompson’s groups T and V , PSL2(Z) and any f.g. group of rational matrices which is bounded. While Symbolic Dynamics [LM95] usually studies subshifts on Z, there has been a lot of work generalizing these results to other groups, from dynamicians and computer scientists working in higher dimensions (Z [Lin04]) to group theorists interested in characterizing group properties in terms of topological or dynamical properties [CSC10]. In this note, we are interested in the existence of aperiodic SFTs, or more generally of aperiodic effective shifts. There has been a lot of work proving how to build aperiodic SFTs in a large class of groups, and more generally tilings on manifolds. The most well known is probably Berger’s construction [Ber64] of an aperiodic SFTs in the two-dimensional lattice Z, but construction on wilder groups or symmetric spaces may be found [Moz97, Coh14]. It is an open question to characterize groups that admit strongly aperiodic SFTs. Cohen[Coh14] showed that f.g. groups admitting strongly aperiodic SFTs are one ended and asked whether it is a sufficient condition. Our first result proves that it is not: If G is finitely presented, then it also must have decidable word problem. This is proven in section 2. This is true more generally for f.g. groups admitting strongly aperiodic effective subshifts, that is subshifts given by a list of forbidden patterns we can enumerate by a program. In fact, we also do not need strongly aperiodic subshifts, but something weaker, that we call weakly strongly aperiodic subshifts: Strongly aperiodic