Weiss's Question

The conjecture has been proved for a number of special classes of amenable groups.

• Weiss [Wei84] and, independently, Slaman and Steel [SS88], proved the conjecture for the group $\mathbb{Z}$.
• Weiss (unpublished) proved the conjecture for groups of the form $\mathbb{Z}^n$.
• Jackson, Kechris and Louveau [JKL02] proved the conjecture for finitely generated groups of polynomial growth.
• Gao and Jackson [GJ15] proved the conjecture for all countable abelian groups. Their methods were extended by Schneider and Seward [SS24] to prove the conjecture for all locally nilpotent groups.
• Conley, Jackson, Marks, Seward, and Tucker-Drob [CJMSTD23] proved the conjecture for all polycyclic groups as well as groups whose finitely generated subgroups all have polynomial growth.

This question was originally asked by Weiss in [Wei84]. Part of Weiss's motivation was a theorem of Ornstein and Weiss which showed that if $E$ is the orbit equivalence relation of a Borel action of a countable amenable group $G$ on a standard Borel space $X$ then for any Borel probability measure $\mu$ on $X$, $E$ is $\mu$-hyperfinite. It follows that a counterexample to Weiss's conjecture would give an example of a countable Borel equivalence relation which is measure hyperfinite, but not hyperfinite.