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arxiv: 2408.12565 · v5 · pith:ZKBJHDKPnew · submitted 2024-08-22 · 🧮 math.DS · math.LO· math.PR

Uniform Borel Amenability

G\'abor Elek , \'Ad\'am Tim\'ar This is my paper

Pith reviewed 2026-05-23 22:09 UTC · model grok-4.3

classification 🧮 math.DS math.LOmath.PR
keywords uniform Borel amenabilityrandomized Borel hyperfinitenessBorel graphscountable Borel equivalence relationshyperfinitenessamenabilityConnes-Feldman-Weiss theoremalmost finiteness
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The pith

Uniform Borel amenability is equivalent to randomized Borel hyperfiniteness for bounded-degree graphs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines uniform Borel amenability as a quantitative strengthening of amenability that must hold uniformly across the entire class of bounded-degree Borel graphs generating countable equivalence relations. It establishes that this property is equivalent to randomized Borel hyperfiniteness, a probabilistic relaxation of the classical hyperfiniteness condition. The equivalence produces three strengthenings of the Connes-Feldman-Weiss theorem. In the Følner setting the same property is also equivalent to randomized Borel almost finiteness, and the graphs become hyperfinite after removal of a single compressible invariant set.

Core claim

A bounded-degree Borel graph is uniformly Borel amenable if and only if it is randomized Borel hyperfinite. For the subclass of uniformly Borel amenable Følner graphs this equivalence further specializes to randomized Borel almost finiteness. Uniformly Borel amenable graphs are hyperfinite modulo a compressible invariant set and satisfy an Ornstein-Weiss-type packing theorem that holds uniformly over every invariant probability measure.

What carries the argument

Uniform Borel amenability, the requirement that a quantitative amenability condition holds simultaneously and uniformly for every bounded-degree Borel graph in the class.

If this is right

  • Three quantitative strengthenings of the Connes-Feldman-Weiss theorem follow directly.
  • Almost finiteness holds outside a single invariant null set for Følner graphs.
  • An Ornstein-Weiss packing theorem holds uniformly over all invariant measures.
  • Hyperfiniteness is recovered after deleting one compressible invariant set.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The uniform control may allow direct transfer of packing and finiteness statements to Borel graphs coming from arbitrary amenable group actions without freeness assumptions.
  • The randomized hyperfiniteness formulation could serve as a template for introducing probabilistic versions of other Borel combinatorial properties.
  • The compressible-set removal result suggests that global hyperfiniteness statements can be recovered by excising a set invisible to every invariant measure.

Load-bearing premise

The amenability condition is required to hold uniformly across the whole class of bounded-degree Borel graphs rather than graph by graph.

What would settle it

A concrete bounded-degree Borel graph generating a countable equivalence relation that meets the uniform amenability definition but admits no sequence of randomized hyperfinite approximations would falsify the claimed equivalence.

read the original abstract

We study a uniform, quantitative form of the amenability-hyperfiniteness paradigm for bounded-degree Borel graphs generating countable Borel equivalence relations. We introduce \emph{uniform Borel amenability} and prove that it is equivalent to \emph{randomized Borel hyperfiniteness}, a probabilistic version of hyperfiniteness. Consequences are three strengthenings of the Connes-Feldman-Weiss theorem. In the setting of uniformly Borel amenable F\o lner graphs (e.g. Borel graphs of not necessarily free actions of amenable groups or Borel graphs of subexponential growth), we establish an analogous equivalence to randomized Borel almost finiteness. We further obtain measure-theoretic structural results, including almost finiteness outside a $\mu$-null invariant set extending a recent result of Conley et al. for free amenable actions, and an Ornstein--Weiss type packing theorem that is uniform over all invariant measures. Finally, we show that uniformly Borel amenable graphs are hyperfinite modulo a compressible invariant set, i.e., after removing a Borel invariant set that is of measure zero for every invariant probability measure.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper introduces uniform Borel amenability for bounded-degree Borel graphs generating countable Borel equivalence relations and proves its equivalence to randomized Borel hyperfiniteness. It derives three strengthenings of the Connes-Feldman-Weiss theorem, an analogous equivalence to randomized Borel almost finiteness for uniformly Borel amenable Følner graphs, measure-theoretic results including almost finiteness outside a μ-null invariant set and a uniform Ornstein-Weiss packing theorem, and hyperfiniteness of such graphs modulo a compressible invariant set.

Significance. If the equivalences and strengthenings hold, the work supplies a uniform quantitative refinement of the amenability-hyperfiniteness correspondence together with probabilistic and measure-theoretic consequences that extend results of Conley et al. The explicit credit for the equivalences, the uniform packing theorem over all invariant measures, and the removal of a compressible null set strengthens the contribution beyond the classical CFW theorem.

minor comments (2)
  1. The abstract states the main equivalences without proof sketches; while the full text presumably supplies them, a brief outline of the key steps in the introduction would improve accessibility.
  2. Notation for the randomized versions (e.g., the probability space underlying randomized hyperfiniteness) should be introduced once with a forward reference to the precise definition in §3.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the detailed summary of its contributions, and the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity; equivalences between new definitions and known theorem

full rationale

The paper defines uniform Borel amenability and randomized Borel hyperfiniteness (plus randomized Borel almost finiteness in the Følner case) as new quantitative strengthenings, then proves their equivalence and derives consequences that strengthen the classical Connes-Feldman-Weiss theorem for bounded-degree Borel graphs generating countable equivalence relations. These steps consist of definitions followed by equivalence proofs and structural results (hyperfiniteness outside a compressible null set, uniform Ornstein-Weiss packing); no equation reduces a claimed prediction to a fitted parameter by construction, no load-bearing premise rests on a self-citation whose content is itself unverified, and no ansatz or uniqueness claim is imported from the authors' prior work. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Pure mathematics paper that introduces definitions and proves equivalences; no numerical parameters are fitted to data and no new entities are postulated beyond standard Borel and measure-theoretic objects.

pith-pipeline@v0.9.0 · 5722 in / 1238 out tokens · 52217 ms · 2026-05-23T22:09:39.446615+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

  1. [1]

    Ab ´ ert., Y

    M. Ab ´ ert., Y. Glasner and B. Vir ´ag, Kesten’s theorem for invariant random subgroups. Duke Math. J. 163 (3), (2014) 465 - 488

    work page 2014

  2. [2]

    Bernshteyn and F

    A. Bernshteyn and F. Weilacher personal communication

    work page

  3. [3]

    Conley, S.C

    C.T. Conley, S.C. Jackson, D. Kerr, A.S. Marks, B.M. Seward a nd R. D. Tucker-Drob, Følner tilings for actions of amenable groups, Math. Annalen , 371, (2018) 663-683

    work page 2018

  4. [4]

    Conley, S.C

    C.T. Conley, S.C. Jackson, A.S. Marks, B.M. Seward and R. D. T ucker- Drob, Borel asymptotic dimension and hyperfinite equivalence relations, Duke Math- ematical Journal , 172(16), (2023) 3175-3226

    work page 2023

  5. [5]

    Connes, J

    A. Connes, J. Feldman and B. Weiss , An amenable equivalence relation is gener- ated by a single transformation, Ergodic Theory and Dynamical Systems , vol. 1 (1981), 431-450

    work page 1981

  6. [6]

    Dougherty, S

    R. Dougherty, S. Jackson and A.S. Kechris , The structure of hyperfinite Borel equivalence relations, Trans. Amer. Math. Soc. , 341(1) (1994), 193-225

    work page 1994

  7. [7]

    Elek and ´A

    G. Elek and ´A. Tim ´ar, Strong almost finiteness, https://arxiv.org/pdf/2308.14554

    work page arXiv

  8. [8]

    Feldman and C.C

    J. Feldman and C.C. Moore , Ergodic equivalence relations and von Neumann algebras, I, Trans. Amer. Math. Soc. , 234 (1977), 289-324

    work page 1977

  9. [9]

    Gromov , Random walk in random groups

    M. Gromov , Random walk in random groups. Geom. Funct. Anal. 13 (2003), no. 1, 73–146

    work page 2003

  10. [10]

    Guentner, N

    E. Guentner, N. Higson and S. Weinberger , The Novikov conjecture for linear groups. Publ. Math. Inst. Hautes ´Etudes Sci. (2005), no. 101, 243–268

    work page 2005

  11. [11]

    Jackson, A.S

    S. Jackson, A.S. Kechris and A. Louveau , Countable Borel equivalence rela- tions,J. Math. Logic , 2(1) (2002), 1-80

    work page 2002

  12. [12]

    Kechris , Amenable versus hyperfinite Borel equivalence relations, J

    A.S. Kechris , Amenable versus hyperfinite Borel equivalence relations, J. Symb. Logic, 58(3) (1993), 894-907

    work page 1993

  13. [13]

    Kechris, S

    A.S. Kechris, S. Solecki and S. Todorcevic , Borel chromatic numbers, Adv. Math., 141 (1999), 1-44

    work page 1999

  14. [14]

    A. S. Kechris , The theory of countable Borel equivalence relations. Cambridge Uni- versity Press , 2024

    work page 2024

  15. [15]

    Marks , A determinacy approach to Borel combinatorics

    A.S. Marks , A determinacy approach to Borel combinatorics. Journal of the Amer. Math. Soc. , 29 (2016), 579-600

    work page 2016

  16. [16]

    Ornstein and B

    D.S. Ornstein and B. Weiss , Entropy and isomorphism theorems for actions of amenable groups. Journal d’Analyse Math´ ematique, 48(1), (1987), 1-141

    work page 1987

  17. [17]

    Osajda , Residually finite non-exact groups

    D. Osajda , Residually finite non-exact groups. Geom. Funct. Anal. 28(2018), no.2, 509–517. 20 G ´ABOR ELEK AND ´AD ´AM TIM ´AR

    work page 2018

  18. [18]

    Roe , Hyperbolic groups have finite asymptotic dimension

    J. Roe , Hyperbolic groups have finite asymptotic dimension. Proc. Amer. Math. Soc. 133 (2005), no.9, 2489–2490

    work page 2005

  19. [19]

    Rokhlin , A ”general” measure-preserving transformation is not mixing

    V. Rokhlin , A ”general” measure-preserving transformation is not mixing. Doklady Akademii Nauk SSSR (N.S.) , 60, (1948), 349-351

    work page 1948

  20. [20]

    Tessera , Quantitative property A, Poincar´ e inequalities, Lp-compression and Lp- distortion for metric measure spaces

    R. Tessera , Quantitative property A, Poincar´ e inequalities, Lp-compression and Lp- distortion for metric measure spaces. Geometriae Dedicata 136 (2008), 203-220

    work page 2008

  21. [21]

    Weiss , Measurable dynamics, Contemp

    B. Weiss , Measurable dynamics, Contemp. Math. , 26 (1984), 395-421

    work page 1984

  22. [22]

    Yu , The coarse Baum-Connes conjecture for spaces which admit a un iform embed- ding into Hilbert space

    G. Yu , The coarse Baum-Connes conjecture for spaces which admit a un iform embed- ding into Hilbert space. Invent. Math. 139 (2000), no. 1, 201-240

    work page 2000

  23. [23]

    R. J. Zimmer , Hyperfinite factors and amenable ergodic actions, Invent. Math. , 41 (1977), 23-31. G´ abor Elek, Department of Mathematics and Statistics, Lancaster Univer- sity, Lancaster, United Kingdom and Alfr´ ed R´ enyi Institute of Mathematics, Budapest, Hungary. g.elek[at]lancaster.ac.uk ´Ad´ am Tim´ ar, Division of Mathematics, The Science Institu...

    work page 1977