Measure-hyperfinite countable Borel equivalence relations and hyperfiniteness

Determine whether every measure-hyperfinite countable Borel equivalence relation is hyperfinite.

Background

The paper situates its study of boundary actions within a longstanding problem in descriptive set theory concerning the relationship between measure-hyperfiniteness and hyperfiniteness for countable Borel equivalence relations. The authors investigate hyperfiniteness for orbit equivalence relations arising from group actions on Roller boundaries, but do not resolve the general measure-hyperfinite-versus-hyperfinite problem.

References

Recently, orbit equivalence relations appearing naturally in geometric group theory started to be explored partially motivated by a fundamental long standing open problem asking whether every measure-hyperfinite countable Borel equivalence relation is hyperfinite (see Section 16.4).

— Borel complexity for product of trees and commuting partial maps  (2609.34068 - Oyakawa, 28 Sep 2026) in Section 1, Introduction