MathDB

Miller–Tserunyan orbit-equivalence question

For every n∈N∪{∞}n\in\mathbb{N}\cup\{\infty\} and every ergodic treeable probability-measure-preserving equivalence relation R\mathcal{R} of cost nn on a standard probability space (X,μ)(X,\mu), does there exist an essentially free probability-measure-preserving action Fn↷(X,μ)\mathbf{F}_n\curvearrowright(X,\mu) whose orbit equivalence relation is R\mathcal{R} and such that, for every g∈Fn∖{e}g\in\mathbf{F}_n\setminus\{e\}, the transformation induced by gg is ergodic?

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Treeable equivalence-relation formulation with total weak mixing

    Every ergodic treeable probability-measure-preserving equivalence relation of cost n∈N∪{∞}n\in\mathbb{N}\cup\{\infty\} is generated by a free totally weak mixing probability-measure-preserving action of Fn\mathbf{F}_n; that is, every nonidentity element acts weakly mixingly.

    source: Wróbel, “Orbit equivalence and total weak mixing of free group actions”

References

Primary source

arXiv

Additional references

ergodic-theorygroup-actions
Posted

Progress summary

Refreshed
Claimed solved

An unrefereed August 2026 paper claims to settle the question affirmatively, but the claim has not yet been independently checked.

The question asks whether an ergodic treeable equivalence relation can be generated by an essentially free action of a free group whose nonidentity elements are all ergodic.

Known results

  • The edge-sliding construction gives an essentially free action of Fn\mathbb{F}_n for every treeable ergodic relation of cost nn, with all standard generators ergodic; the stronger all-elements question is posed there.

August 2026 claimed resolution

A new preprint claims that the required action exists, using a comeager edge-sliding argument, and presents this as settling the Miller–Tserunyan question. It is an unrefereed preprint, with no independent verification or reported objection found in the scan.

Current status (as of August 2026): The question is claimed solved by the preprint, but its proof remains unverified; no independent corroboration was found.

Sources

Solutions 0

No solutions have been posted yet.