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

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 ≥ 2. 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 Z 2 or Goodman-Strauss's on H2.

Strongly Aperiodic SFTs on Generalized Baumslag-Solitar groups

“…The case of groups quasi-isometric to BS(2, 3) is also treated in this paper: in Section 5 we explain how to construct a strongly aperiodic SFT on BS(2, 3). Groups G = BS(1, n) for some n > 1 are already known to possess a minimal strongly aperiodic SFT [AS20]. In total we are able to construct strongly aperiodic SFTs for all GBS.…”

“…5 Adaptation to the Baumslag-Solitar group BS(2, 3) Amenable Baumslag-Solitar groups BS(1, n) are known to have strongly aperiodic SFTs [EM22] and even minimal strongly aperiodic SFTs [AS20]. The case of BS(m, n) for m = n and m, n > 1 has remained unsolved until now.…”

Strongly aperiodic SFTs on generalized Baumslag–Solitar groups

“…The case of groups quasi-isometric to BS (2,3) is also treated in this paper: in §7, we explain how to construct a strongly aperiodic SFT on BS (2,3). Groups G = BS(1, n) for some n > 1 are already known to possess a minimal strongly aperiodic SFT [7]. In total, we are able to construct strongly aperiodic SFTs for all GBS.…”

“…Aperiodicity of the SFT comes from both the aperiodicity of the dynamical system and the clever encoding. These two techniques have successfully been generalized to amenable Baumslag-Solitar groups BS (1, n) [4,7].…”

“…Adaptation to the Baumslag-Solitar group BS(2, 3) Amenable Baumslag-Solitar groups BS (1, n) are known to have strongly aperiodic SFTs [24] and even minimal strongly aperiodic SFTs [7]. The case of BS(m, n) for m = n and m, n > 1 has remained unsolved until now.…”