A strongly aperiodic set of tiles in the hyperbolic plane

Goodman-Strauss, Chaim

doi:10.1007/s00222-004-0384-1

Cited by 29 publications

(27 citation statements)

References 12 publications

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“…De Bruijn's higher dimensional analogue of Sturmian sequences [DB81b,DB81a]. J. Kari gave a third model [Kar96], which was adapted to give the first strongly aperiodic tilings of H n [GS05]. We will give a list of groups known to have strongly aperiodic SFTs momentarily, but first we survey groups known not to have such subshifts.…”

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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“…This is the case for the P-action on the hulls of examples in [8] as well as for the translation group action on the hull of the Euclidean Penrose tiling. The space (T ) is called the continuous hull of T .…”

Section: On Invariant Measures Of Finite Affine Type Tilings 1161mentioning

confidence: 97%

On invariant measures of finite affine type tilings

Petite¹

2006

Ergod. Th. Dynam. Sys.

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In this paper, we consider tilings of the hyperbolic 2-space H 2 , built with a finite number of polygonal tiles, up to affine transformation. To such a tiling T , we associate a space of tilings: the continuous hull (T ) on which the affine group acts. This space (T ) inherits a solenoid structure whose leaves correspond to the orbits of the affine group. First, we prove that the finite harmonic measures of this laminated space correspond to finite invariant measures for the affine group action. Then we give a complete combinatorial description of these finite invariant measures. Finally, we give examples with an arbitrary number of ergodic invariant probability measures.

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“…More examples were discovered by Margulis and Mozes [12]: indeed they discovered a set P consisting of a single tile, a triangle, that admits only nonperiodic tilings. In Goodman-Strauss [8], an example is provided of a finite set P of tiles such that each tiling T 2 T .P / has no nontrivial symmetries. Dranishnikov and Schroeder [6] gave a new construction of aperiodic tile sets.…”

Section: Isommentioning

confidence: 99%