Period-rigidity of one-relator groups
Solène J. Esnay, Ugo Giocanti, Etienne Moutot
TL;DR
The paper addresses when finitely generated groups are periodically rigid by linking aperiodicity properties of subshifts of finite type to group structure. It proves that a group with a presentation $G = <S|R>$ with $|S|\ge 3$ and $R$ cyclically reduced is not periodically rigid, so periodic rigidity within this class occurs only for virtually cyclic groups or torsion-free virtually $Z^2$, confirming a special case of Bitar's conjecture. The authors develop hereditary tools for transferring aperiodicity from subgroups via free and right extensions, apply the Freiheitssatz and Magnus rewriting to one-relator groups, and use MacManus' quasi-planar classification to extend the rigidity dichotomy to a broad class. The results illuminate the deep connections between geometric group structure and symbolic dynamics, including corollaries for surface and quasi-planar groups and a framework for future extensions to more general classes of groups.
Abstract
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.