{"id":"778474a2-b133-4dd3-acdc-eba2e66352ab","arxiv_id":"2608.02327","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Two non-isomorphic ICC property (T) groups Γ1, Γ2 with isomorphic group von Neumann algebras are constructed explicitly, refuting Connes' rigidity conjecture for ICC property (T) groups.","lead":"The paper constructs two explicit infinite discrete groups that are non-isomorphic yet produce isomorphic group von Neumann algebras, and both have Kazhdan's property (T) — disproving Connes' 1982 rigidity conjecture for such groups. The mechanism is a nonlinear fiber shear that makes two different group actions look identical to the operator-algebra construction, while module invariants keep the groups distinct.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The internal argument is coherent; the main load-bearing risk is the unverified import of property (T) for SL3(F2[t]) from EJZK17.","rationale":"I independently scrutinized the internal mathematics. The construction of Γ_i is explicit; the isomorphism L(Γ1)≅L(Γ2) via the fiber shear F(z,y)=(z,y+R(z)) is correct and relies only on standard Fourier/duality for countable abelian groups; the Boolean weight bound in Lemma 4.3 and its use in Proposition 4.4 check out; the relative property (T) proof in Proposition 4.6 is rigorous; the ICC proof and the semisimple/nonsemisimple non-isomorphism argument are coherent. I found no internal error. The single load-bearing vulnerability is the external citation for property (T) of SL3(F2[t]). The reader's weakest assumption identifies exactly this dependency. The concern is not that the cited theorem is false; it is that the manuscript neither quotes its hypotheses nor verifies them for F2[t]. Since every later step in the property (T) chain depends on this, the proof is conditional until that verification is supplied or a self-contained proof is given. The Lean formal-check claim and the provenance of the OpenAI work are secondary and do not affect the mathematical central claim. Therefore the appropriate verdict remains CONDITIONAL, unchanged from the reader's verdict.","tokens_in":16962,"tokens_out":33259,"duration_ms":248742,"concrete_test":"Retrieve the published [EJZK17] and inspect the exact statement of Theorem 1.1 and Section 1.2. Confirm (a) that the theorem applies to the elementary Chevalley group E_Φ(R) for every finitely generated commutative unital ring R and every reduced irreducible root system Φ of rank at least 2; (b) that R=F2[t] is finitely generated as a unital ring (it is, e.g., Z[x]/(2)); and (c) that A2 has rank 2. If all three hold, the property (T) premise of Proposition 4.1 is sound. If the theorem's hypotheses differ, the property (T) of Γ_i, and hence Theorem A, is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem A requires both Γ1 and Γ2 to have property (T). The proof imports this property for SL3(R), R=F2[t], from [EJZK17, Theorem 1.1 and Section 1.2] in Proposition 4.1, without quoting or verifying the theorem's hypotheses. Every subsequent rigidity step depends on this: Proposition 4.6 uses property (T) of SL3(R) to obtain SL3(R)-fixed vectors in an arbitrary almost invariant representation, and Proposition 4.8 uses Lemma 4.7 to lift this to Γ_i. If the cited theorem does not cover EL_3(F2[t])—for example, if it requires a different notion of finite generation, a larger rank, or excludes characteristic 2—then both Γ_i lose property (T) and the counterexample to Connes' rigidity conjecture collapses. In fact, the standard form of EJZK17 states property (T) for elementary Chevalley groups over any finitely generated commutative ring for root systems of rank ≥2, and F2[t] and the A2 root system appear to satisfy those hypotheses, but the manuscript does not supply the verification. This is the weakest load-bearing link.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":17068,"tokens_out":42997,"duration_ms":329896,"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":[{"comment":"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.","section":"Proposition 4.1 / §4"}],"minor_comments":[{"comment":"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.","section":"AI use statement"},{"comment":"In the C-orbit argument, 'Varying g gives P=0' should read 'varying r,s' (or 'varying the pair r,s') for clarity.","section":"Lemma 5.2"},{"comment":"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.","section":"Section 6"},{"comment":"The title contains a typo: 'PROPER TY (T)' should be 'PROPERTY (T)'.","section":"Title"},{"comment":"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.","section":"Proposition 3.4"},{"comment":"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.","section":"Introduction"}],"recommendation":"major_revision","confidential_remarks":"The internal mathematics appears sound and the construction is clever and explicit. My concern is limited to making the property (T) import from [EJZK17] fully explicit and to tidying auxiliary claims. Once the cited theorem's hypotheses are stated and verified for EL3(F2[t]), I expect the paper to be publishable in a strong journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: this paper gives two explicit ICC property (T) groups with isomorphic group von Neumann algebras and non-isomorphic underlying groups, which would kill Connes' original 1982 rigidity conjecture. If the main theorem holds, it's a genuine breakthrough. The construction is concrete — semidirect products of an elementary abelian group by H = SL3(F2[t]) × Sp4(F2) — and the isomorphism of the group factors comes from a quadratic fiber shear on the Pontryagin dual that conjugates the two H-actions. That's new. So is the proof of relative property (T) via Boolean polynomial weight bounds, and the semisimple/nonsemisimple module argument that shows the groups are not isomorphic. I checked the load-bearing computations and they go through. The internal logic is coherent.\n\nThe soft spot is exactly where the reader's report puts it. Proposition 4.1 imports property (T) for SL3(F2[t]) from Ershov–Jaikin-Zapirain–Kassabov without quoting the theorem or verifying its hypotheses. The standard statement does cover elementary Chevalley groups of rank at least two over finitely generated commutative rings, and F2[t] with A2 fits, but the manuscript doesn't say that. This is a missing verification at a load-bearing joint. If the import fails, both groups lose property (T) and the counterexample collapses. The fix is cheap: quote the theorem, check the hypotheses, or prove it directly. A referee should demand that.\n\nTwo smaller items. The paper claims a Lean 4.32.1 formal check of selected parts, but no certificates or scripts are shipped. That claim is unverifiable and should either be backed by artifacts or dropped. And the provenance paragraph about GPT-5.6 Sol and concurrent OpenAI work is not mathematical evidence; it doesn't affect the result but should be handled by editorial policy, not by the authors alone.\n\nBottom line: this deserves a serious referee. The internal case is strong, but I would not accept it until the EJZK import is verified and the Lean claim is either substantiated or removed. If those conditions are met, this is a major result.","headline":"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.","tokens_in":17716,"tokens_out":2468,"would_cite":true,"duration_ms":20765,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46L10","46L55","22D10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Property (T) alone does not determine group von Neumann algebras","keywords":["Kazhdan's property (T)","ICC group","group von Neumann algebra","W*-superrigidity","Connes' rigidity conjecture","semidirect product","crossed product","Fourier transform"],"falsifier":"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.","tokens_in":16670,"feed_emoji":"🧩","tokens_out":8226,"duration_ms":64741,"temperature":0.7,"pith_summary":"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.","feed_headline":"Property (T) alone does not determine group von Neumann algebras","feed_subtitle":"Two explicit ICC property (T) groups are non-isomorphic yet their group von Neumann algebras are isomorphic.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Two property (T) groups, same von Neumann algebra","Non-isomorphic property (T) groups share L(G)","ICC property (T) groups undermine Connes rigidity","Property (T) not enough for isomorphic groups","Counterexample to Connes' rigidity conjecture"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Two property (T) groups, same von Neumann algebra","Non-isomorphic property (T) groups share L(G)","ICC property (T) groups undermine Connes rigidity","Property (T) not enough for isomorphic groups","Counterexample to Connes' rigidity conjecture"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000123,"raw_usage":{"total_tokens":899,"prompt_tokens":669,"completion_tokens":230,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":413,"completion_tokens_details":{"reasoning_tokens":155}},"tokens_in":413,"tokens_out":230,"duration_ms":2421,"temperature":1.0,"reasoning_tokens":155,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T09:20:04.748342+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}