Weakly and Strongly Aperiodic Subshifts of Finite Type on Baumslag-Solitar Groups
arXiv:2004.02534 · doi:10.1016/j.tcs.2022.03.010
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 , but like , 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).
New version with better proofs. Removed a subsection on "shift-similar" substitutions, as the definition was not as interesting as we first thought. Most of the results of this subsection were therefore not that helpful in understanding substitutions that can be "embedded" into BS(1,n)