Wasserstein stability and the nonsingular Borel lifting problem

[2606.03721] Wasserstein stability and the nonsingular Borel lifting problem Skip to main content Search Submit Donate Log in Mathematics > Dynamical Systems arXiv:2606.03721 (math) [Submitted on 2 Jun 2026 ( v1 ), last revised 24 Aug 2026 (this version, v2)] Title: Wasserstein stability and the nonsingular Borel lifting problem Authors: Nachi Avraham-Re'em , George Peterzil View PDF HTML (experimental) 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 $\sigma$-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 that for a compact measure metric group, on average, the total variation of locally distorted density kernels on the group controls their 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. Comments: A simplification of one proof in an appendix. An additional example, suggested by Fabien Hoareau and Emmanuel Roy, of a locally Wasserstein group that is not a Lévy group Subjects: Dynamical Systems (math.DS) ; Functional Analysis (math.FA) MSC classes: 28D15, 22A05, 37A40, 22F10, 22A10, 43A07 Cite as: arXiv:2606.03721 [math.DS] (or arXiv:2606.03721v2 [math.DS] for this version) https://doi.org/10.48550/arXiv.2606.03721 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Nachi Avraham-Re'em [ view email ] [v1] Tue, 2 Jun 2026 14:41:19 UTC (35 KB) [v2] Mon, 24 Aug 2026 18:25:13 UTC (36 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.DS < prev | next > new | recent | 2026-06 Change to browse by: math math.FA References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. 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