Lamplighters admit weakly aperiodic SFTs

TL;DR

This paper proves that lamplighter groups always admit weakly aperiodic subshifts of finite type, providing a counterexample to the conjecture that only virtually cyclic groups lack such subshifts.

Contribution

It demonstrates that lamplighter groups, which are not virtually cyclic, can have weakly aperiodic SFTs, challenging previous conjectures about group classifications.

Findings

Lamplighter groups admit weakly aperiodic SFTs.

Counterexample to the conjecture linking weakly aperiodic SFTs to virtually cyclic groups.

Advances understanding of subshift structures in group actions.

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.