Strongly Aperiodic SFTs on Generalized Baumslag-Solitar groups

TL;DR

This paper demonstrates that all generalized Baumslag-Solitar groups support strongly aperiodic subshifts of finite type, introducing new constructions and techniques that extend previous results to a broader class of groups.

Contribution

It provides the first proof that all GBS groups admit strongly aperiodic SFTs, using novel path-folding techniques and structural theorems.

Findings

All GBS groups admit strongly aperiodic SFTs.

Introduces path-folding technique for constructing aperiodic SFTs.

Achieves minimal strongly aperiodic SFTs on unimodular GBS groups.

Abstract

We look at constructions of aperiodic SFTs on fundamental groups of graph of groups. In particular we prove that all generalized Baumslag-Solitar groups (GBS) admit a strongly aperiodic SFT. Our proof is based on a structural theorem by Whyte and on two constructions of strongly aperiodic SFTs on $\mathbb{F}_n\times \mathbb{Z}$ and $BS(m,n)$ of our own. Our two constructions rely on a path-folding technique that lifts an SFT on $\mathbb{Z}^2$ inside an SFT on $\mathbb{F}_n\times \mathbb{Z}$ or an SFT on the hyperbolic plane inside an SFT on $BS(m,n)$. In the case of $\mathbb{F}_n\times \mathbb{Z}$ the path folding technique also preserves minimality, so that we get minimal strongly aperiodic SFTs on unimodular GBS groups.