Lamplighters admit weakly aperiodic SFTs

2017/10/10 by Cohen, David Bruce
#20F65 #37B50 #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1710.03707

Abstract

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.

Related

• The large scale geometry of strongly aperiodic subshifts of finite type
• Translation-like Actions and Aperiodic Subshifts on Groups
• Aperiodic Subshifts of Finite Type on Groups
• Shifts on the lamplighter group
• Realizability of Subgroups by Subshifts of Finite Type