We prove that the lamplighter group admits strongly aperiodic SFTs, has undecidable tiling problem, and the entropies of its SFTs are exactly the upper semicomputable nonnegative real numbers, and some other results. These results follow from two relatively general simulation theorems, which show that for a large class of effective subshifts on the sea-level subgroup, their induction to the lamplighter group is sofic; and the pullback of every effective Cantor system on the integers admits an SFT cover. We exhibit a concrete strongly aperiodic set with $1488$ tetrahedra. We show that metabelian Baumslag-Solitar groups are intersimulable with lamplighter groups, and thus we obtain the same characterization for their entropies.

翻译：我们证明灯夫群包含强非周期符号转移系统（SFTs），其铺砌问题不可判定，且该群上SFTs的熵恰好是上半可计算非负实数集，并给出其他相关结果。这些结论源自两个具有相当普适性的模拟定理：对于海平面子群上的一类广泛有效子移位，将其诱导至灯夫群后成为sofic子移位；而整数群上每个有效康托尔系统的拉回均存在SFT覆盖。我们构造了一个具体的包含1488个四面体的强非周期集。此外，通过证明亚交换Baumslag-Solitar群与灯夫群具有相互模拟性，我们获得了关于其熵的相同刻画。