Borel structurability on the 2-shift of a countable group

Seward, Brandon; Tucker-Drob, Robin

doi:10.1016/j.apal.2015.07.005

Cited by 36 publications

(50 citation statements)

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“…Immediately after the first version of our paper appeared, using a measurable version of the Local Lemma, Bernhsteyn [5] proved that free Bernoulli subshift admitting an invariant probability measure exists for any countable group. He also noted that this result follows from a deep theorem of Seward and Tucker-Drob [26]. We can actually characterize those uniformly recurrent subgroups Z for which all the Z-proper actions admit invariant probability measures (Theorem 6).…”

Section: Introductionmentioning

confidence: 88%

Uniformly recurrent subgroups and simple C⁎-algebras

Elek

2018

Journal of Functional Analysis

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We study uniformly recurrent subgroups (URS) introduced by Glasner and Weiss [18]. Answering their query we show that any URS Z of a finitely generated group is the stability system of a minimal Z-proper action. We also show that for any sofic U RS Z there is a Z-proper action admitting an invariant measure. We prove that for a U RS Z all Z-proper actions admits an invariant measure if and only if Z is coamenable. In the second part of the paper we study the separable C * -algebras associated to URS's. We prove that if a URS is generic then its C * -algebra is simple. We give various examples of generic URS's with exact and nuclear C *algebras and an example of a URS Z for which the associated simple C * -algebra is not exact and not even locally reflexive, in particular, it admits both a uniformly amenable trace and a nonuniformly amenable trace.

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Section: Introductionmentioning

confidence: 88%

Uniformly recurrent subgroups and simple C⁎-algebras

Elek

2018

Journal of Functional Analysis

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“…Proof. By work of the author and Tucker-Drob (see [52] for free actions; the case of non-free actions is in preparation), there is a G-equivariant class-bijective Borel map φ :…”

Section: Preliminariesmentioning

confidence: 99%

Positive entropy actions of countable groups factor onto Bernoulli shifts

Seward

2019

J. Amer. Math. Soc.

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“…is a G δ subset of 2 Γ , and so we can consider the restriction of the left shift action to it. By [11], for any a and X as in the previous paragraph, there is a Borel Γ-equivariant map X → F (2 Γ ). Therefore studying the graph G(a 0 , S), called the shift graph of Γ, can give us information about all of the graphs G(a, S).…”

Section: Introductionmentioning

confidence: 97%