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arXiv · 2012.11037

Tilings of the hyperbolic plane of substitutive origin as subshifts of finite type on Baumslag-Solitar groups $BS(1,n)$

Nathalie Aubrun·Michael Schraudner
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Abstract

We present a technique to lift some tilings of the discrete hyperbolic plane -- tilings defined by a 1D substitution -- into a zero entropy subshift of finite type (SFT) on non-abelian amenable Baumslag-Solitar groups $BS(1,n)$ for $n\geq2$. For well chosen hyperbolic tilings, this SFT is also aperiodic and minimal. As an application we construct a strongly aperiodic SFT on $BS(1,n)$ with a hierarchical structure, which is an analogue of Robinson's construction on $\mathbb{Z}^2$ or Goodman-Strauss's on $\mathbb{H}_2$.

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Nathalie Aubrun, Michael Schraudner. 2020-12-20. Tilings of the hyperbolic plane of substitutive origin as subshifts of finite type on Baumslag-Solitar groups $BS(1,n)$. https://arxiv.org/abs/2012.11037

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