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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This content will become publicly available on March 14, 2025
Structural, point-free, non-Hausdorff topological realization of Borel groupoid actions
We extend the Becker–Kechris topological realization and change-of-topology theorems for Polish group actions in several directions. For Polish group actions, we prove a single result that implies the original Becker–Kechris theorems, as well as Sami’s and Hjorth’s sharpenings adapted levelwise to the Borel hierarchy; automatic continuity of Borel actions via homeomorphisms and the equivalence of ‘potentially open’ versus ‘orbitwise open’ Borel sets. We also characterize ‘potentially open’ n-ary relations, thus yielding a topological realization theorem for invariant Borel first-order structures. We then generalize to groupoid actions and prove a result subsuming Lupini’s Becker–Kechris-type theorems for open Polish groupoids, newly adapted to the Borel hierarchy, as well as topological realizations of actions on fiberwise topological bundles and bundles of first-order structures.
Our proof method is new even in the classical case of Polish groups and is based entirely on formal algebraic properties of category quantifiers; in particular, we make no use of either metrizability or the strong Choquet game. Consequently, our proofs work equally well in the non-Hausdorff context, for open quasi-Polish groupoids and more generally in the point-free context, for open localic groupoids.
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- Award ID(s):
- 2224709
- PAR ID:
- 10511130
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Forum of Mathematics, Sigma
- Volume:
- 12
- ISSN:
- 2050-5094
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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