Hyper-u-amenablity and Hyperfiniteness of Treeable Equivalence Relations

We introduce the notions of u-amenability and hyper-u-amenability for countable Borel equivalence relations, strong forms of amenability that are implied by hyperfiniteness. We show that treeable, hyper-u-amenable countable Borel equivalence relations are hyperfinite. One of the corollaries that we get is that if a countable Borel equivalence relation is measure-hyperfinite and equal to the orbit equivalence relation of a free continuous action of a virtually free group on a $σ$-compact Polish space, then it is hyperfinite. We also obtain that if a countable Borel equivalence relation is treeable and equal to the orbit equivalence relation of a Borel action of an amenable group on a standard Borel space, or if it is treeable, amenable and Borel bounded, then it is hyperfinite.

💡 Research Summary

The paper introduces two new strengthening notions of amenability for countable Borel equivalence relations (CBERs): u‑amenability and hyper‑u‑amenability. Both concepts are motivated by the long‑standing open problem of whether every amenable CBER is hyperfinite. While hyperfiniteness (the existence of an increasing sequence of finite Borel equivalence relations whose union is the given relation) is known to imply amenability, the converse is unknown in general. The authors focus on treeable relations—those admitting an acyclic Borel graphing—and prove that treeability together with hyper‑u‑amenability forces hyperfiniteness.

Key definitions.
Let ((X,\rho)) be a Borel extended metric space (the metric may take the value (\infty)). A countable Borel equivalence relation (E) on (X) is u‑amenable with respect to (\rho) if there exist Borel maps (\lambda_n:E\to