Abstract It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable.
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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
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1215/00127094-2022-0100
