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

Solène Esnay (1) , Etienne Moutot (2)

1 MC2 - Modèles de calcul, Complexité, Combinatoire
2 CNRS - Centre National de la Recherche Scientifique

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 like 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).

Mots clés: Symbolic dynamics; Aperiodicity; Subshift of finite type; Tilings; Cayley graphs; Substitutions; Groups

Domaines: Mathématique discrète [cs.DM]

Solène Esnay, Etienne Moutot. Weakly and Strongly Aperiodic Subshifts of Finite Type on Baumslag-Solitar Groups. Theoretical Computer Science, 2022, 917, pp.31-50. ⟨10.1016/j.tcs.2022.03.010⟩. ⟨hal-03658172⟩