Lamplighters admit weakly aperiodic SFTs
David Bruce Cohen · Groups Geometry and Dynamics · 2020
Let A be a finite set and G a group. A closed subset X of A^G is called a subshift if the action of G on A^G preserves X . If K is a closed subset of A^G such that membership in K is determined by looking at a fixed finite set of coordinates, and X is the intersection of all translates of K under the action of G , then X is called a subshift of finite type (SFT). If an SFT is nonempty and contains no finite G -orbits, it is said to be weakly aperiodic. A virtually cyclic group has no weakly aperiodic SFT, and Carroll and Penland have conjectured that a group with no weakly aperiodic SFT must be virtually cyclic. Answering a question of Jeandel, we show that lamplighters always admit weakly aperiodic SFTs.