arXiv:2606.03721v1

Wasserstein stability and the nonsingular Borel lifting problem

Nachi Avraham-Re'em, George Peterzil

math.DSmath.FA28D1522A0537A4022F1022A1043A07

Paper metadata

Original submission: June 2, 2026
Claimed MSC: 28D15, 22A05, 37A40, 22F10, 22A10, 43A07
arXiv categories: math.DS, math.FA
Paper license: CC BY 4.0 ↗

Abstract

The Borel lifting problem asks when a nonsingular near action of a Polish group can be represented by a genuine Borel action. For locally compact Polish groups, a classical theorem of Mackey and Ramsay gives an affirmative answer. At the opposite extreme, Glasner-Tsirelson-Weiss proved that every probability preserving Borel action of a Lévy group is trivial, and asked whether a Lévy group can admit a nontrivial nonsingular Borel action. We prove a fixed-point theorem which gives a negative answer for most of the standard Lévy groups in the literature: if $G$ is a locally Wasserstein group, then every quasi-invariant $σ$ -finite Borel measure for a Borel action of $G$ is supported on the fixed points. The proof introduces Wasserstein stability, a local-to-global principle for compact measure metric groups that refines the Gromov-Milman framework for concentration of measure in topological groups. It asserts for a compact measure metric group that the local total variation of the Haar measure tilted by density kernels, controls global Wasserstein distance from the Haar measure. We show that Wasserstein stability implies concentration, and prove a functional inequality using martingales in the spirit of Milman-Schechtman, which is governed by $\ell^{1}$ -sums of martingale increments. This gives a geometric criterion by which most of the standard Lévy groups in the literature are Wasserstein or locally Wasserstein, including $L^{0}$ -groups with compact targets, measure preserving and nonsingular automorphism groups, full groups of amenable equivalence relations, isometry groups of $L^{p}$ -spaces for $p\neq2$ , the unitary group of the hyperfinite $\mathrm{II}_{1}$ -factor, the infinite dimensional unitary and orthogonal groups, the Cameron-Martin affine group, and the isometry group of the Urysohn space.

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Audit summary

Audited against arXiv v1

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

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Generated August 18, 2026

01Statements4 reported findingsCorrect

The fixed-point theorem for nonsingular actions of locally Wasserstein groups, the martingale-chain criterion for Wasserstein stability, and the listed principal examples are supported.

Theorem 1Correct

Nonsingular actions are supported on fixed points

Pages 2 and 11–16 · Sections 5–6 · arXiv:2606.03721v1

For each compact subgroup in a Wasserstein skeleton, averaging the Radon–Nikodym cocycle produces an invariant equivalent probability measure $\nu_n$ . Wasserstein stability makes conditional and Haar averages asymptotically equal on continuous test functions, hence $\nu_n\Rightarrow\mu$ . Invariance passes to the dense skeleton and then to the full group. Repeating this for every measure equivalent to $\mu$ and applying Lemma 6.2 forces the action to be pointwise trivial almost everywhere. Pullback along the homomorphisms defining local Wasserstein generation completes the general case.

Theorem 2Correct

Martingale-chain criterion for Wasserstein groups

Pages 2 and 7–10 · Sections 3–4 · arXiv:2606.03721v1

Successive conditional expectations along the subgroup chain give martingale increments controlled in $L^1$ by the quotient radii. The bounded sum of radii supplies uniform global control, while the largest radius tending to zero turns vanishing local total-variation oscillation into vanishing Wasserstein distance. Corollary 3.5 applies this estimate uniformly to the increasing compact skeleton and yields Theorem 2.

Propositions 2.4 and 2.7Correct

Wasserstein stability is strictly stronger than concentration

Pages 5–7 · Section 2.3 · arXiv:2606.03721v1

Proposition 2.4 constructs, from a set with large complement outside its metric neighborhood, a tilted density whose local total-variation oscillation is small but whose Wasserstein displacement is bounded below; this proves that every Wasserstein family is a Lévy family. For normalized Hamming cubes, Gaussian concentration gives the sharp Lévy threshold $\beta>1/2$ , whereas the martingale chain and the explicit point-mass mixture give the Wasserstein threshold $\beta\geq1$ , so the converse implication genuinely fails at the family level.

Appendices A and BCorrect

Standard Wasserstein and locally Wasserstein groups

Pages 16–24 · arXiv:2606.03721v1

For measurable-map groups, full groups, measure-preserving automorphism groups, and the listed unitary and isometry groups, the finite-dimensional or finite-partition skeletons have uniformly bounded sums of quotient radii and scales tending to zero, so Theorem 2 applies. The locally generated examples in Appendix B are continuous images or finite products of these Wasserstein pieces, exactly as required by Definition 4.3.

02Proofs2 reported findingsCorrect

The functional inequality, Radon–Nikodym averaging, weak-convergence, and fixed-point arguments are correct and complete.

Claims 6.3–6.6Correct and complete

From cocycle averaging to full invariance

Pages 13–16 · arXiv:2606.03721v1

The cocycle identity gives the correct transformation rule for $D_nf$ , hence $\nu_n=D_n\mu$ is $K_n$ -invariant and $Q_nf=D_nf/D_n$ . Strong continuity of the nonsingular Koopman representation makes the local total-variation modulus vanish. Lemma 6.1 then compares $Q_n\phi$ with the Haar average $R_n\phi$ , proving $\nu_n\Rightarrow\mu$ . Closedness of the stabilizer in the continuous action on probability measures extends invariance from the dense union to $G$ .

Proposition 3.3 and Corollary 3.5Correct and complete

Martingale functional inequality

Pages 7–10 · Section 3 · arXiv:2606.03721v1

Conditional expectations along the subgroup chain form orthogonal martingale differences. Borel representatives of each quotient coset lie within its radius $\rho_i$ , giving the $L^\infty$ bound for Lipschitz increments and the matching $L^1$ bound for density increments. Telescoping and Kantorovich duality yield $W_d(\phi m,m)\leq2\sum_i\rho_i\,v_m(\phi,\varrho)$ ; integrating over kernels proves the displayed modulus bound. Uniformly bounded chain length and scale tending to zero then give Wasserstein stability.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2606.03721v1
Authors listed: Nachi Avraham-Re'em, George Peterzil
Audit date: August 18, 2026

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Wasserstein stability and the nonsingular Borel lifting problem

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Nonsingular actions are supported on fixed points

Martingale-chain criterion for Wasserstein groups

Wasserstein stability is strictly stronger than concentration

Standard Wasserstein and locally Wasserstein groups

From cocycle averaging to full invariance

Martingale functional inequality

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