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

TL;DR

It is shown that for residually finite Baumslag-Solitar groups there exist both strongly and weakly-but-not-strongly aperiodic SFTs, and that there exists a strongly a periodicity SFT on BS(n,n), which is similar to that presented by Aubrun and Kari.

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 $\mathbb{Z}^2$, but like $\mathbb{Z}^3$, strong and weak aperiodic SFTs are different classes of SFTs in residually finite BS groups. More precisely, we prove that a weakly aperiodic SFT on BS(m,n) due to Aubrun and Kari is, in fact, strongly aperiodic on BS(1,n); and weakly but not strongly aperiodic on any other BS(m,n). In addition, we exhibit an SFT which is weakly but not strongly aperiodic on BS(1,n); and we show that there exists a strongly aperiodic SFT on BS(n,n).

Figures (7)

• Figure 1: A Wang tile of $BS(m,n)$
• Figure 2: Illustration of the neighbor rules for $BS(2,2)$.
• Figure 3: How a substitution is applied to a biinfinite word: $y=s(x)$ with $s$ a uniform substitution of size $n$
• Figure 4: One tile from the tileset $\tau_{q,I}$.
• Figure 5: Preservation of the oriented arc distance $d_{arc}$ by $r$ and intersection of the arc $\left(r^l \circ \phi(x),r^l \circ \phi(z)\right)$ and the boundary between $A$ and $B$.

Theorems & Definitions (63)

• Proposition 1.1: Meskin residuallyfinite
• Definition 2.1: (Representation by a sequence)
• Definition 2.2: Multiplicative system
• Definition 2.3: (Immortal and periodic points)
• Definition 2.4: (Level)
• Definition 2.5: (Height)
• Definition 2.6: (Multiplying tileset)
• Proposition 2.1: BSdetailed
• Proposition 2.2
• proof