# Lamplighters admit weakly aperiodic SFTs

### David Bruce Cohen

University of Chicago, USA

## 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.

## Cite this article

David Bruce Cohen, Lamplighters admit weakly aperiodic SFTs. Groups Geom. Dyn. 14 (2020), no. 4, pp. 1241–1252

DOI 10.4171/GGD/579