Hyperfiniteness for Roller-boundary component quotients of geometric product-of-trees actions

Determine whether, for a countable group acting properly and cocompactly on a product of trees, the induced orbit equivalence relation on the quotient of the Roller boundary by path-connected components is hyperfinite.

Background

The paper proves that the quotient relation on path-connected components of a Roller boundary need not even be measure-hyperfinite for a proper action on a product of uniformly locally finite trees. The counterexample is not cocompact, motivating the unresolved question of whether proper and cocompact actions have better behavior. The question specifically concerns the quotient ∂RX/GX\partial_R X/G_X, where GXG_X is the median graph on the Roller boundary and the quotient identifies points in the same path-connected component.

References

While the action in Proposition \ref{prop:Roller mod G is not hyperfin} is proper, it is not cocompact. This raises the following question.

\begin{que}\label{que:geometric action} Suppose that a countable group $G$ acts on product of trees $X$ properly and cocompactly. Is $E_G{\partial_R X / G_X}$ hyperfinite? \end{que}

— Borel complexity for product of trees and commuting partial maps  (2609.34068 - Oyakawa, 28 Sep 2026) in Section 2, immediately following Proposition 2.?, “Roller mod G is not hyperfin” (labeled \ref{prop:Roller mod G is not hyperfin}); Question \ref{que:geometric action}