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A Boolean Lévy group with a free nonsingular action

A mix of human-written and AI-generated textHuman understanding: all partsmath.DS — Dynamical Systemsmath.LO — Logic

Contributed by Konstantin Slutsky ↗

Version 1 / Oct 29, 2026 / CC BY 4.0

Abstract

We construct a Boolean Lévy group with a free ergodic spatial nonsingular action on a standard probability space, answering a question of Glasner, Tsirelson and Weiss. The action admits no equivalent invariant σ\sigma-finite measure. The group is a quotient of a weighted summable Boolean group by a discrete subgroup. By the nonsingular fixed-point theorem of Avraham-Re'em and Peterzil, the resulting group is not locally Wasserstein.

Provenance statement

The example presented in this paper was constructed by AI using OpenAI GPT-6-Astra and OpenAI GPT-6.1-sol. These models also assisted with the writing, and Claude Fable-5.1 and Claude Opus-5.5 were used for proofreading. I have substantially revised and edited the draft to improve readability.

Tools used

OpenAI
CodexVersion 6-astra
Anthropic
Claude FableVersion 5.1
Anthropic
Claude OpusVersion 5.5
OpenAI
CodexVersion 6.1-sol

References

  1. N. Avraham-Re'em and G. Peterzil, Wasserstein stability and the nonsingular Borel lifting problem , preprint, arXiv:2606.03721v2 (https://arxiv.org/abs/2606.03721v2), 2026. Version 2, August 24, 2026.arXiv
  2. N. Avraham-Re'em and E. Roy, Poissonian actions of Polish groups , Adv. Math. 479 (2025), article 110437.
  3. M. Gromov and V. D. Milman, A topological application of the isoperimetric inequality , Amer. J. Math. 105 (1983), 843--854.
  4. E. Glasner, B. Tsirelson and B. Weiss, The automorphism group of the Gaussian measure cannot act pointwise , Israel J. Math. 148 (2005), 305--329.
  5. F. Hoareau and F. Le Maître, Spatial models for boolean actions in the infinite measure-preserving setup , Trans. Amer. Math. Soc. 379 (2026), 4263--4305.
  6. S. Kakutani, On equivalence of infinite product measures , Ann. of Math. (2) 49 (1948), 214--224.
  7. A. S. Kechris, Classical Descriptive Set Theory , Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995.
  8. C. C. Moore, Invariant measures on product spaces , in Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability , Vol. II, Part 2: Probability Theory, L. M. Le Cam and J. Neyman (eds.), University of California Press, Berkeley, 1967, 447--459. Berkeley Digital Collections (https://digicoll.lib.berkeley.edu/record/113095/files/math_s5_v2_p2_article-29.pdf).link
  9. V. Pestov, Forty-plus annotated questions about large topological groups , in Open Problems in Topology II , E. M. Pearl (ed.), Elsevier, Amsterdam, 2007, 439--450.
  10. W. A. Veech, Topological dynamics , Bull. Amer. Math. Soc. 83 (1977), 775--830.

Version history

  1. v1Submitted by Konstantin SlutskyInitial depositCurrentOct 29, 2026