A generalization of the simulation theorem for semidirect products

Barbieri, Sebastián; Sablik, Mathieu

doi:10.1017/etds.2018.21

Cited by 15 publications

(17 citation statements)

References 21 publications

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“…More generally, [Jea15b] has shown that polycyclic groups admit strongly aperiodic subshifts of finite type. • Work of Barbieri and Sablik [BS16] shows that any group of the form Z 2 ⋊ H, where H has decidable word problem, admits a strongly aperiodic SFT. This is a very broad collection of groups since it includes Z 2 × H for any H with decidable word problem, as well as the group Sol 3 ∼ = Z 2 ⋊ Z.…”

Section: Introductionmentioning

confidence: 99%

Strongly aperiodic subshifts on surface groups

Cohen¹,

Goodman-Strauss²

2017

Groups Geom. Dyn.

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This paper is devoted to proving the following theorem.Theorem. A hyperbolic group admits a strongly aperiodic subshift of finite type if and only if it has at most one end.We introduce the subject in Section 1 and give an informal outline in Section 2. In Section 3, we formally define our terms and set up the proof, which is a combination of the results of Sections 3-9 as follows:Proof of the Theorem. Propositions 8. 5, 8.12, and 9.5 show that any one-ended hyperbolic group G admits a non-empty subshift of finite type in which no configuration has an infinite order stabilizer. By Proposition 3.3, G admits a subshift of finite type in which no configuration has a stabilizer of finite order. Proposition 3.4 shows that the product of these subshifts is a strongly aperiodic subshift of finite type on G.By Proposition 3.3 every zero-ended group (that is, every finite group) admits a strongly aperiodic subshift of finite type, and [Coh17] shows no group with two or more ends admits such a subshift. 3(4): 220-234, 1960.

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Section: Introductionmentioning

confidence: 99%

Strongly aperiodic subshifts on surface groups

Cohen¹,

Goodman-Strauss²

2017

Groups Geom. Dyn.

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“…Free groups cannot possess strongly aperiodic SFTs [MS85], and a finitely generated and recursively presented group with an aperiodic SFT necessarily has a decidable word problem [Jea15]. Groups that are known to admit a strongly aperiodic SFT are Z 2 [Rob71] and Z d for d > 2 [CK96], fundamental groups of oriented surfaces [CGS17], hyperbolic groups [CGSR17], discrete Heisenberg group [SSU20] and more generally groups that can be written as a semi-direct product G = Z 2 φ H, provided G has decidable word problem [BS19], and amenable Baumslag-Solitar groups [EM20].…”

Section: Introductionmentioning

confidence: 99%

Tilings of the hyperbolic plane of substitutive origin as subshifts of finite type on Baumslag-Solitar groups $BS(1,n)$

Aubrun,

Schraudner

2020

Preprint

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We present a technique to lift some tilings of the discrete hyperbolic plane -tilings defined by a 1D substitution-into a zero entropy subshift of finite type (SFT) on non-abelian amenable Baumslag-Solitar groups BS(1, n) for n ≥ 2. For well chosen hyperbolic tilings, this SFT is also aperiodic and minimal. As an application we construct a strongly aperiodic SFT on BS(1, n) with a hierarchical structure, which is an analogue of Robinson's construction on Z 2 or Goodman-Strauss's on H2.

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“…There has been a recent interest in symbolic dynamics on more general contexts, such as where the lattice Z 2 is replaced by the Cayley graph of an infinite, finitely generated group. Using again the existence of strongly aperiodic SFTs, the domino problem was shown to be undecidable, apart from Z d , on some semisimple Lie groups [18], the Baumslag-Solitar groups [2], the discrete Heisenberg group (announced, [20]), surface groups [10,1], semidirect products on Z 2 [6] or some direct products [4], polycyclic groups [13], some hyperbolic groups [11]. .…”

Section: Introductionmentioning

confidence: 99%

Necessary conditions for tiling finitely generated amenable groups

Menibus¹,

Cornejo²

2020

Discrete &Amp; Continuous Dynamical Systems - A

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We consider a set of necessary conditions which are efficient heuristics for deciding when a set of Wang tiles cannot tile a group.Piantadosi [19] gave a necessary and sufficient condition for the existence of a valid tiling of any free group. This condition is actually necessary for the existence of a valid tiling for an arbitrary finitely generated group.We then consider two other conditions: the first, also given by Piantadosi [19], is a necessary and sufficient condition to decide if a set of Wang tiles gives a strongly periodic tiling of the free group; the second, given by Chazottes et. al. [9], is a necessary condition to decide if a set of Wang tiles gives a tiling of Z 2 .We show that these last two conditions are equivalent. Joining and generalising approaches from both sides, we prove that they are necessary for having a valid tiling of any finitely generated amenable group, confirming a remark of Jeandel [14].

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A generalization of the simulation theorem for semidirect products

Cited by 15 publications

References 21 publications

Strongly aperiodic subshifts on surface groups

Strongly aperiodic subshifts on surface groups

Tilings of the hyperbolic plane of substitutive origin as subshifts of finite type on Baumslag-Solitar groups $BS(1,n)$

Necessary conditions for tiling finitely generated amenable groups

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