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Countable abelian group actions and hyperfinite equivalence relations

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An equivalence relation E on a standard Borel space is hyperfinite if E is the increasing union of countably many Borel equivalence relations \(E_n\) where all \(E_n\)-equivalence classs are finite. In this article we establish the following theorem: if a countable abelian group acts on a standard Borel space in a Borel manner then the orbit equivalence relation is hyperfinite. The proof uses constructions and analysis of Borel marker sets and regions in the space \(2^{{\mathbb {Z}}^{<\omega }}.\) This technique is also applied to a problem of finding Borel chromatic numbers for invariant Borel subspaces of \(2^{{\mathbb {Z}}^n}\).

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Authors and Affiliations

  1. Department of Mathematics, University of North Texas, 1155 Union Circle #311430, Denton, TX, 76203, USA

    Su Gao & Steve Jackson

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  1. Su Gao
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  2. Steve Jackson
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Correspondence to Steve Jackson.

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S. Gao acknowledges United States NSF grants DMS-0501039, DMS-0901853 and DMS-1201290 for the support of his research. S. Jackson acknowledges United States NSF grants DMS-0901853 and DMS-1201290 for the support of his research.

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Gao, S., Jackson, S. Countable abelian group actions and hyperfinite equivalence relations. Invent. math. 201, 309–383 (2015). https://doi.org/10.1007/s00222-015-0603-y

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  • DOI: https://doi.org/10.1007/s00222-015-0603-y

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