Weakly and Strongly Aperiodic Subshifts of Finite Type on Baumslag-Solitar Groups

Abstract

We study the periodicity of subshifts of finite type (SFT) on Baumslag-Solitar groups. We show that for residually finite Baumslag-Solitar groups there exist both strongly and weakly-but-not-strongly aperiodic SFTs. In particular, this shows that unlike Z 2 , but as Z 3 , the notions of strong and weak periodicity are different for residually finite BS groups. More precisely, we prove that the weakly aperiodic SFT on BS(m, n) presented by Aubrun and Kari [3] is in fact strongly aperiodic on BS(1, n). In addition, we exhibit an SFT which is weakly but not strongly aperiodic on BS (1, n). Finally, we show that there exists a strongly aperiodic SFT on BS(n, n).The use of tilings as a computational model was initiated by Wang in the 60s [20] as a tool to study specific classes of logical formulas. His construction consisted in square tiles with colors on each side, that can be placed next to each other if the colors match. Then Wang studied the tilings of the discrete infinite plane (Z 2 ) with these tiles. A similar model exists to tile the infinite line (Z) with two-sided dominoes. Wang quickly realized that a key property was the periodicity of these tilings. It was already known that if a set of dominoes tiled Z, it was always possible to do it in a periodic way, and Wang suspected that it was the same for Wang tiles on Z 2 . However a few years later, one of his students, Berger, proved otherwise by providing a set of Wang tiles tiling the plane but only in aperiodic ways [4]. Alternative aperiodic sets of Wang tiles have been provided by many since then [13,14,18].The study of tilings with Wang tiles was extended to tilings using other symbols with adjacency rules, and became a part of symbolic dynamics, a more general way to encode a smooth dynamical system into symbolic states and trajectories. This discretization approach, see [10] for a comprehensive historiography, is itself a subpart of the field of discrete dynamical systems. In this broader context, it is interesting to study the set of all possible tilings for a given set of symbols and adjacency rules, called a Subshift of Finite Type. It can be equivalently described by the set of Wang tiles producing it, or by a finite alphabet associated with a finite set of forbidden patterns. Z-subshifts (tilings of the integer line) have been studied extensively, and many of their properties are known (see for example [15]). Z d −subshifts

Strongly aperiodic SFTs on hyperbolic groups: where to find them and why we love them

“…In [3] Aubrun and Kari constructed SFTs on BS(1, n) groups, which were shown to be SA by Esnay and Moutot in [16]. Furthermore, in a recent paper Aubrun and Kari [4] showed that the Domino Problem is undecidable for all BS(m, n).…”

Simulations and the lamplighter group

“…These notions are equivalent for Z 2 , but differ in general: On Z 3 , one can build weakly-but-not-strongly aperiodic tilesets easily from strongly aperiodic tilesets of Z 2 , they exist on all Baumslag-Solitar groups which are not virtually abelian [3,11], and Cohen constructs one for the lamplighter group in [8]. On Z 3 and (at least) amenable Baumslag-Solitar groups, also strongly aperiodic tilesets exist, while on L none are known.…”

Subshifts and colorings on ascending HNN-extensions of finitely generated abelian groups

“…These groups were first introduced (though their origin might be older) in [BS62] by G. Baumslag and D. Solitar, in order to provide an example of a non-Hopfian group with two generators and one relator, namely, one which is isomorphic to one of its (proper) quotient groups. Since then these groups have gained attention in the fields of combinatorial group theory and geometric group theory as examples and counterexamples of different properties [Har03,Mes72], and more recently in the field of symbolic dynamics [AK13, CFKP16,EM20].…”