Strongly Aperiodic SFTs on Generalized Baumslag-Solitar 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.