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REVIEW 3 major objections 4 minor 44 references

A continuum of non-measure equivalent groups

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read There exists a continuum of pairwise non-measure-equivalent finitely generated groups with property (T), zero ℓ²-Betti numbers, and no torsion.

desk verdict Genuinely new construction and rigidity results, but the torsion-free 'moreover' is broken and the abstract promises a theorem the body doesn't contain. read the letter →

arxiv 2512.04531 v2 pith:VATD5NY7 submitted 2025-12-04 math.GR math.DSmath.OA

Adrian Ioana , Robin Tucker-Drob This is my paper · ORCID
classification math.GRmath.DSmath.OA MSC 20F6520F6737A2020E22
keywords measureequivalenceproperty(T)ℓ²-Bettinumberswreath-likeproductsproductrigiditycountableBorelrelationsnon-smoothnessclassC
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves that there are continuum many — as many as real numbers — pairwise non-measure-equivalent countable groups that are finitely generated. The groups all have property (T) and are torsion-free, and they have zero ℓ²-Betti numbers in every dimension, so the standard numerical invariants used to separate groups give no information here. The construction encodes each binary sequence into a wreath-like product over a fixed hyperbolic group, and a new rigidity theorem for infinite direct sums decodes the sequence back from the measure equivalence class. The same result settles the question of whether the measure equivalence relation on finitely generated groups is smooth: it is not, and it lies above every countable Borel equivalence relation in the Borel reducibility hierarchy.

What carries the argument

The load-bearing object is the wreath-like product: a group G sitting in an extension 1 → A^(B) → G → B → 1 where conjugation permutes the summands of the core A^(B) according to the action of the quotient B. The paper proves a rigidity theorem (Theorem 6.1) saying that measure equivalence of two such groups passes to measure equivalence of their cores, and a product-rigidity theorem (Theorem 4.4) extending known finite-product rigidity to infinite direct sums of class-C groups. These two results fit together as a decoding chain: G_x ~_ME G_y ⇒ ⊕_{k∈x} A_k^(B) ~_ME ⊕_{k∈y} A_k^(B) ⇒ N_x = N_y ⇒ x=y.

What would settle it

Exhibit a group A in class C whose countable direct sum ⊕_ℕ A is not in class C; if such an A exists, Theorem 4.4 cannot be applied to the groups ⊕_{k∈x} A_k^(B) in the proof of Theorem 1.1, and the stated decoding chain collapses.

Watch

Extended reading notes

Core claim

The central claim is that the measure equivalence class of a specially built group remembers an arbitrary binary sequence. Starting from an infinite family {A_k} of pairwise non-measure-equivalent groups in the class C of groups with a mixing unitary representation and nonzero second bounded cohomology, the authors attach to each x∈{0,1}ᴺ the group G_x, an extension with core ⊕_{k∈x} A_k and a fixed torsion-free hyperbolic quotient B, chosen from the wreath-like product family so that G_x has property (T). If G_x and G_y are measure equivalent, a cocycle-rigidity dichotomy for hyperbolic quotients forces the direct-sum cores ⊕_{k∈x} A_k^(B) and ⊕_{k∈y} A_k^(B) to be measure equivalent; a new

Load-bearing premise

The proof silently assumes that the class C of groups (those with a mixing unitary representation and nonzero second bounded cohomology) is closed under countable direct sums; the final step that decodes x from y depends on this closure, which the paper neither proves nor cites.

Editorial extensions

If this is right

  • There exist continuum many pairwise non-measure-equivalent countable groups that are finitely generated, have property (T), are torsion-free, and have zero ℓ²-Betti numbers in every degree.
  • The measure equivalence relation on finitely generated groups is not smooth; it cannot be classified by real-valued invariants in the usual Borel sense.
  • This relation is above every countable Borel equivalence relation in the Borel reducibility hierarchy, so constructing invariants that separate groups is genuinely hard.
  • The construction provides a new source of groups in the 'trivial invariants' regime, where cost and ℓ²-Betti numbers are silent; such groups may be useful for testing other rigidity phenomena.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The coding trick — using the base hyperbolic group to build property (T) extensions and then recovering the coded set from the core — is likely portable to other equivalence relations in measured group theory, such as orbit equivalence or von Neumann equivalence, wherever a matching product-rigidity theorem exists.
  • One might test whether the same construction can be used to prove non-smoothness (and even universality) of other natural equivalence relations on groups, such as isomorphism or quasi-isometry within restricted classes.
  • Because the family lies in the trivial ℓ²-Betti-number regime, it provides a natural testing ground for whether other invariants (e.g., higher-order cohomology, or the first ℓ²-Betti number after quotients) can distinguish these groups where the standard ones cannot.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper claims Theorem 1.1: there is a continuum-sized family {G_x}_{x∈{0,1}^N} of pairwise non-measure-equivalent ICC countable groups with property (T), zero ℓ²-Betti numbers in all degrees, and torsion-free members. The strategy combines a family {A_k} of pairwise non-ME class-C groups; a wreath-like product construction (Proposition 2.4) producing property (T) groups G_x ∈ WR(D_x, B); a measure-equivalence rigidity theorem for wreath-like products (Theorem 6.1); and an infinite-product ME rigidity theorem (Theorem 4.4). The proof aims to recover x from the ME class of G_x. The abstract also states consequences for the Borel complexity of the ME equivalence relation, but those consequences are not proved in the body.

Significance. If the proof can be completed, Theorem 1.1 is a substantial advance: it provides the first continuum of pairwise non-ME finitely generated groups in the regime where the standard invariants — ℓ²-Betti numbers and cost — are trivial, while also having property (T) and torsion-freeness. The paper contains a substantial and largely self-contained cocycle-rigidity section, and the structural reductions (wreath-like rigidity to products, then product rigidity to the index set) are attractive and potentially powerful. However, the current manuscript has load-bearing gaps: one ingredient is based on an impossible Betti-number computation, and another silently assumes a closure property for class C that is neither proved nor cited.

major comments (3)
  1. [Proposition 5.2, Section 5] The asserted value β_1^{(2)}(B_d)=0 is incompatible with the free-product formula used elsewhere in the paper. For infinite groups, β_1^{(2)}(A*Z)=β_1^{(2)}(A)+β_1^{(2)}(Z)+1=β_1^{(2)}(A)+1. Since C_d is infinite, the correct value is β_1^{(2)}(B_d)=β_1^{(2)}(C_d)+1≥1, not 0. Thus the proof of Proposition 5.2, and the 'can be taken torsion-free' part of Theorem 1.1 that depends on it, fails as written. The cited [26, Théorème 6.8] cannot yield the stated equality. This ingredient needs to be corrected or replaced.
  2. [Section 6, proof of Theorem 1.1 (application of Theorem 4.4)] Theorem 6.1 gives ME between D_x^{(B)} = ⊕_{k∈N_x} A_k^{(N)} and D_y^{(B)} = ⊕_{k∈N_y} A_k^{(N)}. To apply Theorem 4.4, the individual factors must be ICC groups in class C. The text says only 'Since A_k is ICC and in class C', but the factors that actually appear are A_k^{(N)}, the countable direct sum of copies of A_k. ICC is easy, but membership of A_k^{(N)} in class C is not proved or cited, and is not immediate from the definition. Without this closure property, the conclusion that the index sets N_x and N_y coincide does not follow from Theorem 4.4 as stated. A lemma establishing this closure, or a reformulation avoiding it, is required.
  3. [Abstract vs. body] The abstract announces two additional results: that the finitely generated ME equivalence relation ≃^{fg}_{ME} is non-smooth, and that it lies above every countable Borel equivalence relation in the Borel reducibility hierarchy. Neither statement appears in the introduction or body, and Theorem 1.1 alone does not imply them. The manuscript should either prove these claims or amend the abstract to state only what is established.
minor comments (4)
  1. [Proposition 5.2 / [26, Théorème 6.8]] Please recheck the cited reference for the vanishing β_1(B_d)=0. The cited theorem more naturally gives the free-product formula, which would contradict the claimed value.
  2. [Proposition 5.1] The calculation β_1(A_c)=c uses β_1(F_2×F_2)=0 together with the +1 term from the free-product formula. A sentence noting this would prevent confusion, since F_2×F_2 itself has vanishing first ℓ²-Betti number.
  3. [Section 6, final paragraph] The appeal to 'the Künneth formula' for the infinite direct sum D_x^{(B)} should be justified by a reference. Künneth formulas are usually stated for finite direct products, and the infinite direct sum case needs a separate argument.
  4. [Corollary 3.2] In the proof, 'Claim (1)' should likely be a reference to Lemma 3.3(1) rather than to an internal claim.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular reduction in the main derivation; the rigidity chain is logically independent of the conclusion. Self-citations are load-bearing but not circular.

full rationale

The main chain is: G_x ~_ME G_y ⇒ (Theorem 6.1) D_x^(B) ~_ME D_y^(B) ⇒ (Theorem 4.4) the index sets N_x and N_y coincide ⇒ x = y. This does not assume the conclusion: x is encoded combinatorially and recovered by injectivity/rigidity. The rigidity theorems (3.1, 4.4, 6.1) are proved in the paper by cocycle arguments; they are not renamed versions of the target result. The only external inputs are standard ME invariants (Gaboriau cost, ℓ²-Betti), Monod-Shalom class C facts, and the authors' prior wreath-like-product existence theorems [13,14]. Those prior theorems assert existence of property (T) groups in WR(-,B) under assumptions that do not include pairwise non-ME of the G_x, so they are not a re-labelling of the continuum theorem. There is no fitted parameter renamed as a prediction. Two non-circular correctness caveats should be weighed separately: (a) Proposition 5.2 asserts β_1^(2)(C_d * Z)=0 via [26], while the standard free-product formula used elsewhere gives β_1^(2)(C_d)+1 ≥ 1, undermining the 'can be taken torsion-free' assertion if correct; (b) applying Theorem 4.4 to the factors A_k^(N) needs each such infinite direct sum to be in class C, while the text only says 'Since A_k is ICC and in class C', silently assuming a closure property. These are gaps/errors, not circular reductions, so they do not raise the circularity score.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The paper contributes the rigidity theorems (3.1, 4.4, 6.1) and the combination strategy; it imports the property (T) wreath-like product existence engine ([13, 14]) and the separation tools ([25, 26, 39]) from prior work. No fitted parameters, no new postulated entities. The main unstated burden is the closure of class C under countable direct sums, needed in the last step of Theorem 1.1.

assumptions (9)
  • domain assumption Property (T) wreath-like products exist over arbitrary direct sums of n-generated groups (Proposition 2.4, via [14, Prop 2.11, Lemmas 2.8/2.10/2.12] and [13, Lemma 3.22, Cor 4.18, Prop 4.21])
    Load-bearing: produces G_x ∈ WR(D_x, B) with property (T). Entirely delegated to the authors' prior work; [13] is to appear at submission.
  • standard math Monod–Shalom [39, Theorem 1.3]: certain free products lie in class C
    Used in Props 5.1/5.2 to place A_c and B_d in class C.
  • standard math Gaboriau's cost/ℓ²-Betti and approximate ergodic dimension machinery [25, 26]
    Separates A_c by β_1 = c and C_d by a-erg-dim = d.
  • standard math Brothier–Deprez–Vaes [10, Theorem 4.1] property (S) cocycle dichotomy (Theorem 3.7)
    First step of the hyperbolic cocycle rigidity Theorem 3.1; proof sketched as an adaptation.
  • standard math Adams [2, 3, 4] and Hjorth–Kechris [29]: boundary amenability of hyperbolic groups; H-equivariant maps P^{≥3}(∂H) → F(H); minimal c-equivariant maps
    Used in Section 3 (Cor 3.5, Prop 3.6) to upgrade cocycle amenability to elementarity.
  • standard math Zimmer cocycle reduction [44, Lemma 5.2.11] applied measurably across ergodic components
    Core step of Lemma 4.1 (cocycle reduction to the normal subgroup).
  • domain assumption Class C is closed under countable direct sums (factors A_k^(N) or D_x lie in class C)
    Silently required in the final application of Theorem 4.4 in the proof of Theorem 1.1; neither proved nor cited.
  • standard math Lück [37, Theorem 7.2(6)]: vanishing of ℓ²-Betti numbers in extensions with β-acyclic kernel and an infinite-order element in the quotient; Künneth-type vanishing for direct sums
    Gives β_p(G_x) = 0 at the end of the proof of Theorem 1.1.
  • standard math Alvarez–Gaboriau [5, Theorem 1.5] and the free-product formula for ℓ²-Betti numbers
    Prop 5.2's pairwise separation claim rests on [5, Thm 1.5]; the paper's own β_1(A_c) = c computation shows the operative formula is β_1(A*B) = β_1(A)+β_1(B)+1, making the asserted β_1(B_d) = 0 inconsistent.

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Pith. "Pith review of A continuum of non-measure equivalent groups." pith.science (2026). https://pith.science/paper/VATD5NY7

@misc{pith2026251204531,
  author       = {Pith},
  title        = {Pith review of: A continuum of non-measure equivalent groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VATD5NY7}},
  note         = {Machine review of arXiv:2512.04531}
}
abstract

We construct a continuum sized family $\{G_x\}_{x\in\{0,1\}^{\mathbb N}}$ of pairwise non-measure equivalent countable groups which have property (T) (hence are finitely generated), have zero $\ell^2$-Betti numbers of all orders, and are torsion-free. We also prove that the equivalence relation $\simeq^{\mathrm{fg}}_{\mathrm{ME}}$ of measure equivalence between finitely generated groups is non-smooth, resolving a question of S. Thomas. Our proof moreover shows that $\simeq^{\mathrm{fg}}_{\mathrm{ME}}$ sits above every countable Borel equivalence relation in the Borel reducibility hierarchy.

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