We follow in this paper a recent line of work, consisting in characterizing the periodically rigid finitely generated groups, i.e., the groups for which every subshift of finite type which is weakly aperiodic is also strongly aperiodic. In particular, we show that every finitely generated group admitting a presentation with one reduced relator and at least $3$ generators is periodically rigid if and only if it is either virtually cyclic or torsion-free virtually $\mathbb Z^2$. This proves a special case of a recent conjecture of Bitar (2024). We moreover prove that period rigidity is preserved under taking subgroups of finite indices. Using a recent theorem of MacManus (2023), we derive from our results that Bitar's conjecture holds in groups whose Cayley graphs are quasi-isometric to planar graphs.

翻译：本文延续了近期一系列研究工作的思路，旨在刻画具有周期刚性的有限生成群，即满足以下性质的群：每个弱非周期的有限型子移位同时也是强非周期的。特别地，我们证明了每个具有至少$3$个生成元且呈现为一个简约关系子的有限生成群，当且仅当其为虚拟循环群或无挠虚拟$\mathbb Z^2$群时，才具有周期刚性。这证明了Bitar（2024）近期猜想的一个特例。此外，我们证明了周期刚性在取有限指数子群下保持不变。利用MacManus（2023）的最新定理，我们从结果中推导出Bitar猜想在凯莱图拟等距于平面图的群中成立。