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We identify natural conditions for a countable group acting on a countable tree which imply that the orbit equivalence relation of the induced action on the Gromov boundary is Borel hyperfinite. Examples of this condition include acylindrical actions. We also identify a natural weakening of the aforementioned conditions that implies measure hyperfiniteness of the boundary action. We then document examples of group actions on trees whose boundary action is not hyperfinite.
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Borel asymptotic dimension and hyperfinite equivalence relations
A long-standing open problem in the theory of hyperfinite equivalence relations asks if the orbit equivalence relation generated by a Borel action of a countable amenable group is hyperfinite. In this paper we prove that this question always has a positive answer when the acting group is polycyclic, and we obtain a positive answer for all free actions of a large class of solvable groups including the Baumslag–Solitar group BS(1, 2) and the lamplighter group Z2 ≀ Z. This marks the first time that a group of exponential volume-growth has been verified to have this property. In obtaining this result we introduce a new tool for studying Borel equivalence relations by extending Gromov’s notion of asymptotic dimension to the Borel setting. We show that countable Borel equivalence relations of finite Borel asymptotic dimension are hyperfinite, and more generally we prove under a mild compatibility assumption that increasing unions of such equivalence relations are hyperfinite. As part of our main theorem, we prove for a large class of solvable groups that all of their free Borel actions have finite Borel asymptotic dimension (and finite dynamic asymptotic dimension in the case of a continuous action on a zero dimensional space). We also provide applications to Borel chromatic numbers, Borel and continuous Følner tilings, topological dynamics, and C∗-algebras.
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- PAR ID:
- 10521280
- Publisher / Repository:
- Duke Mathematical Journal
- Date Published:
- Journal Name:
- Duke Mathematical Journal
- Volume:
- 172
- Issue:
- 16
- ISSN:
- 0012-7094
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1215/00127094-2022-0100
