Aperiodic SFTs on Baumslag-Solitar groups

Esnay, Solène J.; Moutot, Etienne

doi:10.1016/j.tcs.2022.03.010

Cited by 6 publications

(8 citation statements)

References 21 publications

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“…In Strongly aperiodic SFTs on generalized Baumslag-Solitar groups 1227 particular, both torus knot groups and BS(n, n) admit this kind of subshift, as they are unimodular. This latter result improves [24]. COROLLARY 6.6.…”

Section: N Aubrun Et Almentioning

confidence: 65%

“…Amenable Baumslag–Solitar groups are known to have strongly aperiodic SFTs [24] and even minimal strongly aperiodic SFTs [7]. The case of for and has remained unsolved until now.…”

Section: Adaptation To the Baumslag–solitar Groupmentioning

confidence: 99%

“…Free groups and more generally groups with at least two ends cannot possess a strongly aperiodic SFT [17] and a finitely generated and recursively presented group with an aperiodic SFT necessarily has a decidable word problem [32]. Groups that are known to admit a strongly aperiodic SFT are Z 2 [46] and Z d for d > 2 [21], fundamental groups of oriented surfaces [18], 1-ended hyperbolic groups [19], discrete Heisenberg group [48] and more generally groups that can be written as a semi-direct product G = Z 2 φ H -provided G has decidable word problem [9]-, self-simulable groups with decidable word problem [10]-which include braid groups and some RAAGs-and residually finite Baumslag-Solitar groups [24]. Combining results in [17,32], one may conjecture that finitely generated groups admitting a strongly aperiodic SFT are exactly 1-ended groups with decidable word problem.…”

Section: Introductionmentioning

confidence: 99%

“…Adaptation to the Baumslag-Solitar group BS(2, 3) Amenable Baumslag-Solitar groups BS (1, n) are known to have strongly aperiodic SFTs [24] and even minimal strongly aperiodic SFTs [7]. The case of BS(m, n) for m = n and m, n > 1 has remained unsolved until now.…”

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confidence: 99%

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Strongly aperiodic SFTs on generalized Baumslag–Solitar groups

Aubrun¹,

Bitar²,

Huriot-Tattegrain³

2023

Ergod. Th. Dynam. Sys.

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We look at constructions of aperiodic subshifts of finite type (SFTs) on fundamental groups of graph of groups. In particular, we prove that all generalized Baumslag-Solitar groups (GBS) admit a strongly aperiodic SFT. Our proof is based on a structural theorem by Whyte and on two constructions of strongly aperiodic SFTs on $\mathbb {F}_n\times \mathbb {Z}$ and $BS(m,n)$ of our own. Our two constructions rely on a path-folding technique that lifts an SFT on $\mathbb {Z}^2$ inside an SFT on $\mathbb {F}_n\times \mathbb {Z}$ or an SFT on the hyperbolic plane inside an SFT on $BS(m,n)$ . In the case of $\mathbb {F}_n\times \mathbb {Z}$ , the path folding technique also preserves minimality, so that we get minimal strongly aperiodic SFTs on unimodular GBS groups.

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Section: N Aubrun Et Almentioning

confidence: 65%

“…Amenable Baumslag–Solitar groups are known to have strongly aperiodic SFTs [24] and even minimal strongly aperiodic SFTs [7]. The case of for and has remained unsolved until now.…”

Section: Adaptation To the Baumslag–solitar Groupmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Strongly aperiodic SFTs on generalized Baumslag–Solitar groups

Aubrun¹,

Bitar²,

Huriot-Tattegrain³

2023

Ergod. Th. Dynam. Sys.

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“…Within the groups that satisfy these constraints, several are known to admit SA SFTs. For instance polycyclic-by-finite groups which are not virtually cyclic [3,14], one-ended word-hyperbolic groups [10,11], some Baumslag-Solitar groups [1,2,12], groups of the form…”

mentioning

confidence: 99%

Aperiodic subshifts of finite type on groups which are not finitely generated

Barbieri¹

2022

Preprint

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

Eduardo

2021

Ergod. Th. Dynam. Sys.

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For an ascending HNN-extension $G*_{\psi }$ of a finitely generated abelian group G, we study how a synchronization between the geometry of the group and weak periodicity of a configuration in $\mathcal {A}^{G*_{\psi }}$ forces global constraints on it, as well as in subshifts containing it. A particular case are Baumslag–Solitar groups $\mathrm {BS}(1,N)$ , $N\ge 2$ , for which our results imply that a $\mathrm {BS}(1,N)$ -subshift of finite type which contains a configuration with period $a^{N^\ell }\!, \ell \ge 0$ , must contain a strongly periodic configuration with monochromatic $\mathbb {Z}$ -sections. Then we study proper n-colorings, $n\ge 3$ , of the (right) Cayley graph of $\mathrm {BS}(1,N)$ , estimating the entropy of the associated subshift together with its mixing properties. We prove that $\mathrm {BS}(1,N)$ admits a frozen n-coloring if and only if $n=3$ . We finally suggest generalizations of the latter results to n-colorings of ascending HNN-extensions of finitely generated abelian groups.

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Aperiodic SFTs on Baumslag-Solitar groups

Cited by 6 publications

References 21 publications

Strongly aperiodic SFTs on generalized Baumslag–Solitar groups

Strongly aperiodic SFTs on generalized Baumslag–Solitar groups

Aperiodic subshifts of finite type on groups which are not finitely generated

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

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