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.

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Lamplighters承认非周期性SFT

假设$ A $为有限集，$ G $为一组。如果$ G $对$ A ^ G $的操作保留$ X $，则$ A ^ G $的闭合子集$ X $称为子移位。如果$ K $是$ A ^ G $的封闭子集，则$ K $的隶属关系是通过查看固定的有限坐标集确定的，而$ X $是操作下$ K $的所有平移的交点$ G $，则$ X $被称为有限类型的子移位（SFT）。如果SFT是非空的并且不包含有限的$ G $轨道，那么它被称为弱非周期性的。几乎是周期性的组没有弱非周期性SFT，而Carroll和Penland推测没有弱非周期性SFT的组必须是实质上周期性的。回答了让德尔的问题，我们证明了打火机总是接受弱非周期性的SFT。