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Strongly aperiodic SFTs on generalized Baumslag–Solitar groups
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 44 / Issue 5 / May 2024
- Published online by Cambridge University Press:
- 20 June 2023, pp. 1209-1238
- Print publication:
- May 2024
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- Article
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We look at constructions of aperiodic subshifts of finite type (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.
