Group Colorings and Bernoulli Subflows

Gao, Su; Jackson, Steve; Seward, Brandon

doi:10.1090/memo/1141

Cited by 29 publications

(52 citation statements)

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“…Clemens [2] showed that the topological conjugacy relation on subshifts over a finite alphabet is a universal countable Borel equivalence relation. Gao, Jackson, and Seward [11] partially analyzed the topological conjugacy relation on minimal subshifts and proved that this relation is not smooth, i.e. it is strictly more complex than the equality relation ∆ R .…”

Section: Introductionmentioning

confidence: 99%

The complexity of topological conjugacy of pointed Cantor minimal systems

Kaya

2017

Arch. Math. Logic

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In this paper, we analyze the complexity of topological conjugacy of pointed Cantor minimal systems from the point of view of descriptive set theory. We prove that the topological conjugacy relation on pointed Cantor minimal systems is Borel bireducible with the Borel equivalence relation ∆ +Moreover, we show that ∆ + R is a lower bound for the Borel complexity of topological conjugacy of Cantor minimal systems. Finally, we interpret our results in terms of properly ordered Bratteli diagrams and discuss some applications.2010 Mathematics Subject Classification. Primary 03E15, Secondary 37B05.

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Section: Introductionmentioning

confidence: 99%

The complexity of topological conjugacy of pointed Cantor minimal systems

Kaya

2017

Arch. Math. Logic

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“…(However, for subshifts, there is still a group action present that can be exploited: see Section 4. ) The recent monograph of Gao, Jackson, and Seward [9] studies the complexity of topological conjugacy of free, minimal subshifts. (In fact, before their construction, it was an open problem whether such subshifts necessarily exist for every countable G.) It follows essentially from a classical result of Curtis, Hedlund and Lyndon (see [19] or [9, Lemma 9.2.1]) that for any countable group G, topological conjugacy of G-subshifts is a countable Borel equivalence relation.…”

Section: Introductionmentioning

confidence: 99%

“…(In fact, before their construction, it was an open problem whether such subshifts necessarily exist for every countable G.) It follows essentially from a classical result of Curtis, Hedlund and Lyndon (see [19] or [9, Lemma 9.2.1]) that for any countable group G, topological conjugacy of G-subshifts is a countable Borel equivalence relation. Gao, Jackson and Seward showed [9,Corollary 1.5.4] that if G is infinite, this equivalence relation is not smooth, and that if G is locally finite, then it is hyperfinite [9,Theorem 1.5.6]. They also pose the general question [9,Problem 9.4.11] to determine the complexity of this equivalence relation for an arbitrary countable group G. This was an important motivating question for our work and in Theorem 1.2, we provide a partial answer.…”

Section: Introductionmentioning

confidence: 99%

“…Gao, Jackson and Seward showed [9,Corollary 1.5.4] that if G is infinite, this equivalence relation is not smooth, and that if G is locally finite, then it is hyperfinite [9,Theorem 1.5.6]. They also pose the general question [9,Problem 9.4.11] to determine the complexity of this equivalence relation for an arbitrary countable group G. This was an important motivating question for our work and in Theorem 1.2, we provide a partial answer.…”

Section: Introductionmentioning

confidence: 99%

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On the complexity of topological conjugacy of Toeplitz subshifts

Sabok

Tsankov

2017

Isr. J. Math.

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Abstract. In this paper, we study the descriptive set theoretic complexity of the equivalence relation of conjugacy of Toeplitz subshifts of a residually finite group G. On the one hand, we show that if G = Z, then topological conjugacy on Toeplitz subshifts with separated holes is amenable. In contrast, if G is non-amenable, then conjugacy of Toeplitz G-subshifts is a non-amenable equivalence relation. The results were motivated by a general question, asked by Gao, Jackson and Seward, about the complexity of conjugacy for minimal, free subshifts of countable groups.

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“…Seward and Tucker-Drob [ST16] further developed the techniques of [GJS09;GJS16] in order to establish the following very strong result: If Γ ñ X is a free Borel action of Γ on a standard Borel space X, then there exists an equivariant Borel map π : X Ñ 2 Γ such that πpXq is a free subshift [ST16, Theorem 1.1]. (Here, and in what follows, a horizontal line indicates topological closure.)…”

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confidence: 99%

Building large free subshifts using the Local Lemma

Bernshteyn

2019

Groups Geom. Dyn.

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A b s t r ac t . Gao, Jackson, and Seward [GJS09] proved that every countably infinite group Γ admits a nonempty free subshift X Ď 2 Γ . Here we strengthen this result by showing that free subshifts can be "large" in various senses. Specifically, we prove that for any k ě 2 and h ă log 2 k, there exists a free subshift X Ď k Γ of Hausdorff dimension and, if Γ is sofic, entropy at least h, answering two questions attributed by Gao, Jackson, and Seward to Juan Souto [GJS16, Problems 11.2.4 and 11.2.5]. Furthermore, we establish a general lower bound on the largest "size" of a free subshift X 1 contained inside a given subshift X. A central role in our arguments is played by the Lovász Local Lemma, an important tool in probabilistic combinatorics, whose relevance to the problem of finding free subshifts was first recognized by Aubrun, Barbieri, and Thomassé [ABT16].

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Group Colorings and Bernoulli Subflows

Cited by 29 publications

References 0 publications

The complexity of topological conjugacy of pointed Cantor minimal systems

The complexity of topological conjugacy of pointed Cantor minimal systems

On the complexity of topological conjugacy of Toeplitz subshifts

Building large free subshifts using the Local Lemma

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