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Property (T) alone does not determine group von Neumann algebras

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2026-08-04 09:20 UTC pith:Y25GWFVE

load-bearing objection A serious, mostly coherent counterexample to Connes' rigidity conjecture for ICC property (T) groups, with one load-bearing external import (property (T) for SL3(F2[t])) that should be verified before acceptance. the 1 major comments →

arxiv 2608.02327 v1 pith:Y25GWFVE submitted 2026-08-03 math.OA math.DSmath.GR

ICC property(T) groups without W^*-superrigidity

Shuoxing Zhou This is my paper
classification math.OA math.DSmath.GR MSC 46L1046L5522D10
keywords Kazhdan's property (T)ICC groupgroup von Neumann algebraW*-superrigidityConnes' rigidity conjecturesemidirect productcrossed productFourier transform
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper sets out to show that Kazhdan's property (T), together with the infinite-conjugacy-class (ICC) condition, does not force a countable discrete group to be encoded in its group von Neumann algebra. It constructs two explicit groups, Γ1 and Γ2, that are both ICC and have property (T); the groups are not isomorphic, yet their group von Neumann algebras L(Γ1) and L(Γ2) are *-isomorphic. If the construction is correct, it refutes Connes' rigidity conjecture as stated for arbitrary ICC property (T) groups. The proof works by keeping the same semidirect product ingredients and changing only one action, then showing the two resulting crossed-product algebras are isomorphic through a measure-preserving conjugacy on the dual group.

Core claim

The paper's central claim is that the two semidirect products Γ_i = D ⋊_{θ_i} (SL_3(F_2[t]) × Sp_4(F_2)), i=1,2, satisfy all four parts of Theorem A. The abelian kernel D is the same in both groups; only the action θ_i differs, by a 1-cocycle coming from a quadratic refinement of a symplectic form. Fourier transform identifies L(Γ_i) with the crossed product L∞(D̂_i) ⋊ H, and the paper exhibits a Haar-measure-preserving homeomorphism F of the dual that conjugates the two H-actions, giving L(Γ1) ≅ L(Γ2). The groups are distinguished by their Q-module structure: D is semisimple under θ1 and nonsemisimple under θ2, and this distinction is shown to be invariant under isomorphism, so Γ1 ≇ Γ2.

What carries the argument

The isomorphism part rests on the fiber shear F(z,y) = (z, y + R(z)) on the Pontryagin dual of D, where R is a quadratic map built from the symplectic refinement r0(a1,b1,a2,b2)=a1b1+a2b2. F is not a compact-group automorphism, but it is a Haar-preserving homeomorphism and strictly conjugates the two dual actions, which is exactly what makes the two crossed products isomorphic. The non-isomorphism part rests on the Q-module E_ℓ = V* ⊕ k with action twisted by the cocycle ℓ_q = q·r0 − r0; this module appears as a nonsplit extension in one group and is absent as a direct summand in the other, leading to a semisimple-versus-nonsemisimple obstruction.

Load-bearing premise

The construction inherits property (T) of SL_3(F_2[t]) entirely from an external theorem about elementary groups over finitely generated commutative rings; if that theorem does not cover F_2[t], neither group has property (T) and Theorem A collapses.

What would settle it

Look up the cited memoir's main theorem and check whether its hypotheses include the ring F_2[t]; if the theorem excludes non-finite fields or this particular ring, Proposition 4.1 loses its justification and the counterexample fails.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • Connes' rigidity conjecture for arbitrary ICC property (T) groups is false.
  • Kazhdan's property (T), even combined with the ICC condition, does not imply W*-superrigidity.
  • Isomorphisms of group von Neumann algebras can arise from measure-preserving conjugacies on the dual of an abelian subgroup rather than from any group isomorphism.
  • Property (T) groups without W*-superrigidity exist in explicit, countable, discrete form, so rigidity results for higher-rank lattices do not extend to the full class of property (T) groups.

Where Pith is reading between the lines

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

  • The same fiber-shear construction may generalize: other quadratic Boolean refinements or other Euclidean domains could yield additional non-isomorphic ICC property (T) pairs with isomorphic factors, possibly even infinite families.
  • The proof shows that the group factor is sensitive to the orbit structure of the action on the dual, so one could search for new von Neumann algebra invariants that distinguish actions up to measure-preserving conjugacy rather than up to group isomorphism.
  • If the external property (T) input were verified directly for the ring F_2[t], the counterexample would become self-contained; a careful check of the cited theorem's hypotheses is the most direct way to test the construction's validity.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 6 minor

Summary. The paper constructs two explicit countable discrete groups Γ1 = D1 ⋊ (SL3(F2[t]) × Sp4(F2)) and Γ2 = D2 ⋊ (SL3(F2[t]) × Sp4(F2)) with the same underlying F2-vector space D and the same quotient H; only the action of the finite factor Sp4(F2) on D is changed by a 1-cocycle. It proves that both Γi have Kazhdan's property (T), both are ICC, Γ1 and Γ2 are not isomorphic as groups, and L(Γ1) ≅ L(Γ2) as von Neumann algebras. The von Neumann algebra isomorphism is obtained by Fourier transform and a measure-preserving, H-equivariant 'quadratic shear' between the induced actions on the duals; the non-isomorphism is detected by a semisimple-versus-nonsemisimple distinction of the characteristic normal subgroup D as a module over Sp4(F2). The paper concludes that this gives a counterexample to Connes' rigidity conjecture for ICC property (T) groups.

Significance. If correct, this is a major result: it disproves a long-standing conjecture in the negative and shows that property (T) together with ICC does not imply W*-superrigidity. The construction is explicit and essentially parameter-free. The internal derivations are mostly self-contained and checkable: the cocycle identity, the sheared conjugacy, the Boolean-polynomial weight bound, the invariant-measure estimate, and the module-theoretic semisimplicity argument all appear sound. The main external input is the Ershov–Jaikin-Zapirain–Kassabov theorem for property (T) of SL3(F2[t]); this is a standard tool, but its hypotheses are not stated or verified in the manuscript.

major comments (1)
  1. [Proposition 4.1 / §4] The property (T) of SL3(F2[t]) is imported from [EJZK17, Theorem 1.1 and Section 1.2] without stating the theorem's hypotheses. Since Proposition 4.6 and Proposition 4.8 both depend on this fact, the authors should quote the exact form of the theorem and explicitly verify that it applies here: (a) F2[t] is a finitely generated commutative ring; (b) EL3(F2[t]) is the elementary Chevalley group of type A2, whose root system has rank 2; (c) the theorem has no hidden characteristic or rank restriction. I believe the cited theorem does cover this case, but the verification must appear in the paper; if the theorem does not cover EL3(F2[t]), both Γi lose property (T) and the main theorem collapses.
minor comments (6)
  1. [AI use statement] The statement that 'Lean 4.32.1 was used to formally check selected parts of the argument' is not accompanied by any formal statement, files, or theorem names. Either provide the checked statements or remove the claim, as it is not independently verifiable from the manuscript.
  2. [Lemma 5.2] In the C-orbit argument, 'Varying g gives P=0' should read 'varying r,s' (or 'varying the pair r,s') for clarity.
  3. [Section 6] The definition of 'elementary abelian group' as an abelian group in which all non-identity elements have the same order is nonstandard; the usual definition is that all non-identity elements have prime order p for a fixed p. The proof only uses exponent two, so this can be stated in the standard way.
  4. [Title] The title contains a typo: 'PROPER TY (T)' should be 'PROPERTY (T)'.
  5. [Proposition 3.4] The identification L(Di ⋊ H) ≅ L∞(D̂i) ⋊ H is classical, but the action of H on L∞(D̂i) should be specified precisely and a reference (e.g., Takesaki or a standard crossed-product text) would help the reader.
  6. [Introduction] The discussion of independent concurrent work with OpenAI is not needed for the mathematical argument; consider moving it to a footnote or to the acknowledgments so as not to distract from the proof.

Circularity Check

0 steps flagged

No significant circularity: the construction and proofs are genuine derivations; external imports are not self-referential.

full rationale

The derivation chain in Theorem A is not circular. The two groups are explicitly built as semidirect products Γ_i = D_i ⋊ (SL_3(R) × Sp_4(k)) in Section 2. The von Neumann algebra isomorphism in Proposition 3.4 uses the classical Fourier identification L(D_i ⋊ H) ≅ L^∞(D̂_i) ⋊ H and an explicit Haar-measure-preserving homeomorphism F(z,y) = (z, y + R(z)) that is shown in Proposition 3.2 to conjugate the two actions. This is a genuine construction, not a renamed input or fitted prediction. Property (T) is imported from the external published theorem [EJZK17] in Proposition 4.1; although the hypotheses are not verified in detail, this is a correctness/verification concern, not circularity, and the citation is not self-citation. The ICC and non-isomorphism results are proved by internal algebraic arguments (Lemmas 5.1–6.4) that do not presuppose the main theorem. There are no fitted parameters, no data-driven predictions, and no load-bearing self-citations. The AI-assistance and concurrent-work statements are descriptive and are not used as evidence for any mathematical claim.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

No free parameters. The constant 12 in Propositions 4.4 and Corollary 4.5 is a universal combinatorial bound, not fitted; every ingredient of the actions θ_i is explicitly defined with no adjustable constants. No invented physical entities. The groups Γ_i, the modules D_i, the cocycle ℓ_q, and the fiber shear F are constructions with proofs of their properties, not postulates pulled in to force the result. The rubric's 'graviton problem' does not apply to a pure-math existence construction.

axioms (6)
  • standard math EL3(R) has Kazhdan's property (T) for R = F2[t] (via Ershov–Jaikin-Zapirain–Kassabov, cited in Prop 4.1)
    Imported from [EJZK17] without re-derivation or statement of hypotheses; the entire Section 4 property (T) chain rests on it.
  • standard math If N ◁ G, (G,N) relative property (T), and G/N property (T), then G has property (T) ([BR95] Prop 1.3)
    Used twice in Proposition 4.8.
  • standard math Fourier transform identification L(D ⋊ H) ≅ L∞(D̂, μ) ⋊ H for discrete abelian D
    Classical; the engine of Proposition 3.4.
  • standard math Dual of A⊗V* is Hom_k(A,V); symplectic form identifies V ≅ V*
    Section 3 pairing; V ≅ V* is Q-equivariant.
  • standard math Sp4(F2) acts transitively on V\{0}; natural 4-dim module is simple and faithful
    Lemma 6.2 and Proposition 6.5 depend on it.
  • domain assumption Connes' rigidity conjecture in the broad form stated in the introduction is the intended target
    The refutation applies to the stated broad conjecture; the refined higher-rank-lattice version (Houdayer) is explicitly not addressed.

pith-pipeline@v1.3.0-daily-deepseek · 16387 in / 63555 out tokens · 493614 ms · 2026-08-04T09:20:04.748342+00:00 · methodology

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Cite this review

Pith. "Pith review of ICC property(T) groups without W$^*$-superrigidity." pith.science (2026). https://pith.science/paper/Y25GWFVE

@misc{pith2026260802327,
  author       = {Pith},
  title        = {Pith review of: ICC property(T) groups without W$^*$-superrigidity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y25GWFVE}},
  note         = {Machine review of arXiv:2608.02327}
}
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read the original abstract

We construct two explicit countable discrete groups $\Gamma_1$ and $\Gamma_2$ that are both ICC and have Kazhdan's property (T). Although $\Gamma_1$ and $\Gamma_2$ are not isomorphic as groups, their group von Neumann algebras are isomorphic: $L(\Gamma_1)\cong L(\Gamma_2)$. This provides a counterexample to Connes' rigidity conjecture for ICC property (T) groups. This result was obtained with assistance from GPT-5.6 Sol, independently of and concurrently with work by OpenAI.

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Reference graph

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