Period-rigidity of one-relator groups

TL;DR

This paper characterizes when certain one-relator groups are periodically rigid, showing they are either virtually cyclic or virtually Z^2, and confirms a conjecture for groups with planar Cayley graphs.

Contribution

It proves a specific case of Bitar's conjecture by characterizing periodic rigidity in one-relator groups with at least three generators.

Findings

Groups with one relator and ≥3 generators are periodically rigid iff they are virtually cyclic or virtually Z^2.

Period rigidity is preserved under finite index subgroups.

Bitar's conjecture holds for groups with Cayley graphs quasi-isometric to planar graphs.

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.