Different groups with the same operator-algebra shadow
Does a group’s von Neumann algebra uniquely reveal the group that made it?
A good abstraction should forget irrelevant detail without forgetting the object. Connes’s rigidity conjecture proposed that, for a particularly rigid class of groups, the group von Neumann algebra got that balance exactly right: same operator algebra would force the same group.4Connes The new manuscript constructs countably infinitely many different rigid groups with exactly the same factor.1manuscript If correct, this is not a small exception. It identifies the information the abstraction loses.
A concrete starting point
A group is a set of transformations equipped with a precise rule for composition. Rotations of a square, permutations, and addition modulo a fixed number are standard examples.4Connes
A group von Neumann algebra, written L(G), represents the group’s transformations as operators on a Hilbert space and closes them under analytic limits. This retains extensive operator-algebraic structure without obviously retaining the original multiplication table. Connes asked whether the strong conditions ICC and property (T) make that representation complete.1manuscript4Connes
The counterexample starts with the same four labels—00, 01, 10, 11—and two ways to add them. Without carrying, every nonzero label returns to 00 after two additions. With binary carry, 01 goes 00 → 01 → 10 → 11 → 00. Same four equally likely points; different group law. The proof scales this tiny trick into infinite rigid groups while preserving the measured action their factors see.1manuscript2walkthrough
Different group laws, isomorphic operator factors
Switch between two group structures. Their internal multiplication pattern changes, but the measured action sent into the factor stays the same.
What Connes expected the factor to remember
For a countable discrete group G, its group von Neumann algebra L(G) is built from the operators that shift square-summable functions on G. When G is ICC—every nonidentity element has infinitely many conjugates—L(G) is a II₁ factor, a clean and highly structured operator algebra.1manuscript
Property (T) is a different kind of rigidity: any unitary representation with vectors that are almost fixed must contain a genuinely fixed vector. Connes and Jones established the corresponding operator-algebraic property-(T) framework in 1985.5Connes The conjecture asked:
L(G) ≅ L(H), with G and H countable ICC property-(T) groups ⇒ G ≅ H.4Connes
This was never a claim about all groups. In the amenable ICC world, Connes’s classification already implies that many different groups produce the same hyperfinite II₁ factor.1manuscript The hope was that property (T) would prevent that collapse.
What earlier approaches could—and could not—do
Deformation/rigidity theory produced powerful positive reconstruction theorems, and explicit W*-superrigid groups show that some groups really are determined by their bare factors.6Ioana7Chifan Those results concern special constructions. They never proved that ICC plus property (T), by itself, was sufficient for every group.
Several natural counterexample routes hit one of the two hypotheses. Starting with nonisomorphic finite groups that have the same complex group algebra introduces a finite normal subgroup, which breaks ICC. Spreading the finite difference across infinitely many “lamp” coordinates creates almost-invariant vectors and breaks property (T). Naive pushouts or root-adjoining leave small central conjugacy classes or fail to intertwine the acting action.2walkthrough
There was a second benchmark beyond Connes’s yes/no question. Popa proved that a factor can arise from at most countably many ICC property-(T) groups, and later asked whether every such fiber might at least be finite. A two-group counterexample would refute Connes; an infinite family sharing one factor would also make Popa’s countability ceiling sharp.8Popa1manuscript
The opening: a crossed product can forget the addition law
Suppose an abelian group A is acted on by another group K. Fourier duality rewrites the factor of the semidirect product A ⋊ K as a crossed product built from the compact dual Â, its Haar probability measure, and the K-action. That construction can remember how K moves the points while forgetting how those same points add to one another.1manuscript2walkthrough
This is the conceptual pivot: do not search for two isomorphic K-modules. Put two different K-compatible compact group laws on the same measured K-space. If their discrete duals retain a visible algebraic difference, the factors can agree while the groups do not.2walkthrough
The quadratic Boolean module that makes carry equivariant
The acting group K must simultaneously be torsion-free, ICC, property (T), and able to act on characteristic-two data. The manuscript chooses a finite-index subgroup of SL₄(ℤ[t]) whose reduction modulo 2 surjects onto SL₄(𝔽₂[t]); the separate congruence condition modulo 3 supplies torsion-freeness.1manuscript2walkthrough This separation is essential: binary algebra can do the construction, while K itself cannot hide the order-four signature used later.
Set V = 𝔽₂[t]⁴ and b(v) = v ⊗ v. The space B spanned by all b(v) is the divided square of V. In characteristic two, its polarization is
b(u + v) − b(u) − b(v) = u ⊗ v + v ⊗ u.
Functionals q on B therefore evaluate q(b(v)) as Boolean polynomials of degree at most two: diagonal tensor coordinates supply linear terms and off-diagonal coordinates supply quadratic terms.1manuscript This particular quadratic module does two jobs. It packages binary carry in a K-equivariant cocycle, and it later gives the uniform support bound needed for property (T).
Carry changes torsion without changing the measured action
On X × Y = V* × B*, compare coordinatewise addition with
(ℓ,q) ⋆ (ℓ′,q′) = (ℓ + ℓ′, q + q′ + rℓ,ℓ′),
where rℓ,ℓ′(b(v)) = ℓ(v)ℓ′(v). The extra term is exactly binary carry. The resulting map to the carry group C₀ is K-equivariant and measure-preserving, because its translations are triangular: first translate X, then translate each Y-fiber.1manuscript It is not a group isomorphism for the old coordinatewise law.
Dualizing produces two abelian groups. The split group D = V ⊕ B has exponent two: doubling any element gives zero. The carry dual E₀ contains an element of order four. Form Λ = D ⋊ K and Γ₀ = E₀ ⋊ K. Fourier transform and the common measured K-action give L(Λ) ≅ L(Γ₀), while torsion distinguishes Λ from Γ₀ because K itself is torsion-free.1manuscript2walkthrough
Same Haar probability space + same K-action ⇒ same crossed-product factor. Exponent-two torsion versus an order-four element ⇒ different groups.1manuscript
The proof map
- Build the rigid actor. Construct a torsion-free ICC property-(T) group K that still surjects onto SL₄(𝔽₂[t]), so the module computation can happen in characteristic two.1manuscript
- Put two laws on one probability space. Use the divided-square module to globalize the four-point carry as a continuous, symmetric, K-equivariant cocycle. The identity of the underlying points preserves Haar measure and the K-action.1manuscript
- Dualize and separate. Fourier duality identifies the factors. Order-four torsion in the nonsplit carry extension, absent from the split extension, proves the groups are nonisomorphic.1manuscript
- Verify ICC and property (T). Infinite K-orbits on nonzero module elements give ICC. A Boolean quadratic is nonzero on at least one quarter of a finite cube; after excluding the asymptotic one-eighth of nonprimitive vectors, a 1/7 detection estimate supplies the needed relative property-(T) gap.1manuscript2walkthrough
- Shift the carry. Start the carry after n polynomial coefficients. Every n gives the same measured action, while the finite-orbit part of a characteristic 2-torsion quotient has size 24n, recovering n from the abstract group.1manuscript
Technical layer · why the spectral estimate is 1/7
In the finite box VN ≅ 𝔽₂4N, primitive vectors—those whose four polynomial coordinates have greatest common divisor 1—have exact count 7·24N−3 + 1. Their density tends to 7/8.1manuscript
A nonzero Boolean polynomial of degree at most two is nonzero on at least 1/4 of the cube. At worst, every nonprimitive vector lies inside that support; nonprimitive density tends to 1/8. Therefore at least 1/4 − 1/8 = 1/8 of the whole cube consists of primitive detecting vectors. Relative to the 7/8 primitive population, the detection proportion is (1/8)/(7/8) = 1/7.1manuscript
Rank four is not cosmetic. In rank j, the same expression is (1/4 − 21−j)/(1 − 21−j), which becomes positive exactly when j ≥ 4.1manuscript2walkthrough This uniform gap proves relative property (T) for the abelian-by-K construction; property (T) of K then gives property (T) for Λ.
The exact theorem—and its limits
The manuscript’s Theorem 1.2 constructs finitely generated ICC property-(T) groups Λ, Γ₀, Γ₁, … that are pairwise nonisomorphic and satisfy L(Γn) ≅ L(Λ) for every n ≥ 0.1manuscript Each Γn contains a subgroup isomorphic to Γ₀ of index 24n, so the Γ-family is mutually commensurable.1manuscript
The pair Λ and Γ₀ already refutes Connes’s general reconstruction conjecture. The countably infinite family also answers Popa’s finite-to-one question negatively and attains the pre-existing countability bound.1manuscript8Popa The released Comparator challenge names both the two-group and infinite-family statements as formal targets.1manuscript8Popa3formal artifact
This does not say that no group can be recovered from its factor. Specific W*-superrigid property-(T) groups remain valid positive examples.7Chifan It does not settle reconstruction for higher-rank lattices themselves, classify all fibers, or make operator algebras useless. It shows that ICC plus property (T), without additional structure, is insufficient.1manuscript
Assessment of significance
If the theorem withstands independent expert review, yes. It directly answers a problem recorded by Connes in 1994 and produces the strongest possible fiber cardinality compatible with Popa’s countability theorem.4Connes8Popa The construction is also explanatory: the failure comes from a precise measurable-versus-algebraic blind spot, not a black-box existence argument.
The released Lean development is unusually extensive, and the Comparator manifest targets the headline existence theorems.3formal artifact OpenAI also describes a multi-stage model, researcher, formalization, and review process.9announcement That raises confidence; it does not create independent consensus. Priority, correspondence between the formal model and standard operator-algebra definitions, and the analytic inputs deserve close specialist scrutiny.
Is the claim overhyped?
My calibrated verdict: a genuine conjecture-level counterexample if independently validated; the theorem is narrower—and more interesting—than ‘rigidity is dead.’
The construction gives infinitely many pairwise nonisomorphic ICC property-(T) groups with one group factor.
That is the manuscript’s main theorem, and both the basic and infinite-family existence statements are explicit Comparator targets.1manuscript3formal artifact
The same example makes Popa’s countability bound sharp.
The family is countably infinite, while the earlier theorem bounds such property-(T) fibers by countable size.1manuscript8Popa
No rigid group can be reconstructed from its factor.
False. W*-superrigid property-(T) examples still exist. The counterexample shows only that ICC and property (T) are not sufficient hypotheses by themselves.7Chifan1manuscript
Formalization means no independent mathematical review is needed.
The released artifacts materially improve auditability, but experts still need to examine definitions, imported analytic assumptions, theorem-to-headline correspondence, and historical priority.3formal artifact9announcement
Full bibliography
9 fully annotated sources
- 01 · primary manuscript
OpenAI. Ten Advances in Mathematics and Theoretical Computer Science — Chapter 4, abstract, Theorem 1.2, and §§3–6, PDF pp. 96–112. Primary source for the construction, exact theorem, Boolean support estimate, and infinite family. ↩
- 02 · reasoning walkthrough
OpenAI. Reasoning Walkthroughs — Chapter 4, §§4.1–4.8, PDF pp. 20–24. Explains the failed finite and wreath-product routes, binary carry, the 1/7 estimate, and shifted-carry invariant. ↩
- 03 · formal certificate
OpenAI. ConnesRigidity.lean and Comparator challenge — exists_nonisomorphic_propertyT_icc_groups_with_isomorphic_factors and exists_infinite_pairwise_nonisomorphic_propertyT_icc_groups_with_isomorphic_factors; commit e62211d. OpenAI-authored Lean targets from the same release for the basic counterexample and infinite family, under the Comparator permitted-axiom list; not independent review. ↩
- 04 · authoritative reference
Alain Connes. Noncommutative Geometry — Chapter V, Appendix B, Problem 1, p. 551 (1994). The explicit reconstruction problem now called Connes’s rigidity conjecture. ↩
- 05 · peer-reviewed predecessor
Alain Connes and Vaughan F. R. Jones. Property T for von Neumann algebras — Bulletin of the London Mathematical Society 17(1), 57–62 (1985). Primary source relating property (T) of ICC groups and their group factors. ↩
- 06 · peer-reviewed predecessor
Adrian Ioana, Sorin Popa, and Stefaan Vaes. A class of superrigid group von Neumann algebras — Annals of Mathematics 178(1), 231–286 (2013), especially Proposition 3.5. Positive W*-superrigidity context and an alternative proof of countability of property-(T) fibers. ↩
- 07 · peer-reviewed predecessor
Ionut Chifan, Adrian Ioana, Denis Osin, and Bin Sun. Wreath-like products of groups and their von Neumann algebras I: W*-superrigidity — Annals of Mathematics 198(3), 1261–1303 (2023). Property-(T) groups that remain positively W*-superrigid; these are not contradicted by the counterexample. ↩
- 08 · peer-reviewed predecessor
Sorin Popa. Deformation and rigidity for group actions and von Neumann algebras — Proceedings of the ICM Madrid, Vol. I, 445–477 (2007), §4, pp. 457–458. Primary source for the countable-to-one upper bound in the property-(T) setting. ↩
- 09 · official announcement
OpenAI. Ten advances in mathematics and theoretical computer science — “How the results were developed” and “Verification” sections. Used only for OpenAI’s description of research and verification process, not theorem correctness. ↩