We study the problem of realizing families of subgroups as the set of stabilizers of configurations from a subshift of finite type (SFT). This problem generalizes both the existence of strongly and weakly aperiodic SFTs. We show that a finitely generated normal subgroup is realizable if and only if the quotient by the subgroup admits a strongly aperiodic SFT. We also show that if a subgroup is realizable, its subgroup membership problem must be decidable. The article also contains the introduction of periodically rigid groups, which are groups for which every weakly aperiodic subshift of finite type is strongly aperiodic. We conjecture that the only finitely generated periodically rigid groups are virtually $\mathbb{Z}$ groups and torsion-free virtually $\mathbb{Z}^2$ groups. Finally, we show virtually nilpotent and polycyclic groups satisfy the conjecture.

翻译：我们研究将子群族实现为有限型子位移（SFT）构型稳定子集的问题。该问题同时推广了强非周期与弱非周期SFT的存在性问题。我们证明：一个有限生成正规子群可实现的充要条件是该子群的商群容许强非周期SFT。同时证明：若某子群可实现，则其子群成员判定问题必须是可判定的。本文还引入了周期刚性群的概念，即所有弱非周期有限型子位移均为强非周期的群。我们猜想：仅有的有限生成周期刚性群是虚拟$\mathbb{Z}$群以及无挠虚拟$\mathbb{Z}^2$群。最后，我们验证了虚拟幂零群与多循环群均满足该猜想。