Strongly Aperiodic SFTs on Generalized Baumslag-Solitar groups

Nathalie Aubrun (1) , Nicolás Bitar (1) , Sacha Huriot-Tattegrain (2)

1 GALaC - Graphes, Algorithmes et Combinatoire - LISN
2 ENS Paris Saclay - Ecole Normale Supérieure Paris-Saclay

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 first a structural theorem by Whyte and second 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.

Domaines: Systèmes dynamiques [math.DS]; Mathématique discrète [cs.DM]

Nathalie Aubrun, Nicolás Bitar, Sacha Huriot-Tattegrain. Strongly Aperiodic SFTs on Generalized Baumslag-Solitar groups. 2022. ⟨hal-03649781⟩