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

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 \(\mathbb{Z}^3\), s...

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Bibliographic Details
Published in:arXiv.org 2024-10
Main Authors: Esnay, Solène J, Moutot, Etienne
Format: Article
Language:English
Subjects:
Periodic variations
Online Access:Get full text
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Summary: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).
ISSN:2331-8422
DOI:10.48550/arxiv.2004.02534