Strongly aperiodic SFTs on generalized Baumslag–Solitar groups

Abstract

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.

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

“…Groups that are known to admit a strongly aperiodic SFT are Z 2 [21] and Z d for d > 2 [10], fundamental groups of oriented surfaces [8], hyperbolic groups [9], discrete Heisenberg group [22] and more generally groups that can be written as a semi-direct product G = Z 2 ⋊ φ H , provided G has decidable word problem [7], and amenable Baumslag-Solitar groups [11]. In [3] the authors adapt the construction of [4,5] to get a strongly aperiodic SFT on non-amenable Baumslag-Solitar groups BS(m, n). None of these two constructions are, to the best of our knowledge, minimal SFTs.…”

“…In [14] the author adapts [17] to an expansive setting, the trinary tiling, akin to BS (1,3). Later in [4,5] the example of aperiodic SFT that is given is a construction adapted from [17].…”