index

OpenAI’s collection Ten Advances in Mathematics and Theoretical Computer Science reports results obtained by an internal model across ten separate areas.¹ Two of its chapters land in the same corner of mathematics, infinite discrete groups and the von Neumann algebras built from them, and both resolve questions that had been open for decades in the negative direction.

Chapter 3 constructs an explicit group that admits no finite permutation approximations at all. Chapter 4 constructs infinitely many pairwise nonisomorphic rigid groups that are indistinguishable to the operator algebra built from them.

The two are worth reading together because they attack the same instinct from opposite sides: the belief that infinite groups can be pinned down by finite or analytic shadows.

Soficity is approximation by permutations

Let $H$ be a countable group. A sofic approximation of $H$ is a sequence of maps

$$p_n : H \longrightarrow \operatorname{Sym}(Y_n)$$

into the symmetric groups of finite sets $Y_n$, with $|Y_n| \to \infty$. These maps are not required to be homomorphisms. They are required only to look like homomorphisms in a statistical sense, measured by normalized Hamming distance:

$$d_H(p, q) = \frac{\lvert \{ z \in Y : pz \neq qz \} \rvert}{\lvert Y \rvert} \qquad (p, q \in \operatorname{Sym}(Y)).$$

This is the fraction of points on which two permutations disagree. The conditions are:

$$p_n(1) = 1, \qquad d_H\bigl(p_n(gh),\, p_n(g)p_n(h)\bigr) \to 0, \qquad d_H\bigl(p_n(g),\, 1\bigr) \to 1 \;\; (g \neq 1).$$

The second condition says multiplication holds at almost every point of $Y_n$. The third says every nonidentity element of $H$ moves almost every point: the approximation stays faithful, and does not quietly send distinct group elements to nearly the same permutation. A group is sofic if such a sequence exists.

Condition 1: multiplicationp(gh) agrees with p(g)p(h) at almost every pointThe disagreement set shrinks to a vanishing fraction of the model set.

Condition 2: faithfulnessEvery g ≠ 1 moves almost every pointNonidentity elements must not be approximated by near-identity permutations.

model sizemultiplication errordisplacement

|Y| = 12

|Y| = 10³

|Y| = 10⁶

limit

Schematic budgets. A group is sofic when some sequence drives the left column to 0 and the right column to 1 for every fixed pair of elements.

Gromov introduced the property in work on symbolic dynamics,² and Weiss named sofic groups and asked whether a nonsofic one exists. The question hardened into the soficity conjecture: is every countable group sofic?³

Nobody could produce a counterexample, and the class kept absorbing everything thrown at it. Amenable groups are sofic. Residually finite groups, those where every nonidentity element survives in some finite quotient, are sofic. So is the wider LEF class: a group is locally embeddable into finite groups if for every finite subset $F$ there is a finite group $B$ and an injection $\varphi : F \to B$ with

$$\varphi(xy) = \varphi(x)\varphi(y) \qquad (x, y, xy \in F),$$

so each finite chunk of the multiplication table embeds exactly, with $B$ allowed to depend on $F$. Soficity is the approximate version of the same idea, and the class is closed under enough operations that the supply of candidate counterexamples was thin.

all countable groups

sofic

amenableZ, solvable groups

LEF

residually finitefree groups, SL_n(Z)

exact finite chunks

closed under products, extensions by amenable groups, limits · BS(2,3) lives here

not soficL_F₂(1,2)^×unit group of the binary Leavitt algebra

Amenable

Has an invariant finitely additive probability measure. Sofic by averaging Følner sets.

Residually finite

Every nonidentity element survives in some finite quotient.

LEF

Every finite chunk of the multiplication table embeds exactly into a finite group.

Sofic

Every finite chunk embeds approximately, into a symmetric group, in normalized Hamming distance.

Outside

Thompson's group V is finitely presented, infinite and simple, hence not LEF. Soficity of the Leavitt unit group would have forced V to be LEF.

Why anyone cared

Soficity mattered because a long list of conjectures had been proved for sofic groups and remained open in general. That made “every group is sofic” the load-bearing assumption in a whole shelf of theorems.

Gottschalk’s surjunctivity conjecture is the original motivation. A group $H$ is surjunctive if for every finite alphabet $A$, every injective $H$-equivariant continuous map $A^H \to A^H$ is surjective, a cellular-automaton analogue of the fact that an injective self-map of a finite set is onto. Gromov and Weiss proved every sofic group is surjunctive.

Connes’s embedding conjecture for groups asks whether every discrete group is hyperlinear: approximable by finite-dimensional unitary matrices in normalized Hilbert–Schmidt distance. Permutation matrices satisfy

$$\lVert P_\sigma - P_\tau \rVert_{2,n}^2 = 2\, d_H(\sigma, \tau),$$

so every sofic group is hyperlinear. Kaplansky’s direct-finiteness conjecture, which asks whether $ab = 1$ in a group algebra forces $ba = 1$, together with Lück’s determinant conjecture, the algebraic eigenvalue conjecture, and the Kervaire–Laudenbach conjecture all have “true for sofic groups” theorems attached.

Earlier work had produced conditional routes to a nonsofic group: Bowen and Burton from flexible permutation stability of $\mathrm{PSL}_d(\mathbb Z)$, Gohla and Thom from central extensions of $p$-adic lattices under a stability hypothesis. Every route needed an unproved stability assumption. Separately, the Aldous–Lyons conjecture, a broader statement about unimodular random rooted networks of which the soficity conjecture is the restriction to point masses on normal subgroups, was disproved by Bowen, Chapman, Lubotzky and Vidick using machinery descended from $\mathrm{MIP}^* = \mathrm{RE}$. That did not settle soficity, because a non-co-sofic invariant random subgroup need not be supported on a normal subgroup.

The counterexample

Let $\mathbb F_2$ be the field with two elements. The binary Leavitt algebra is

$$R = L_{\mathbb F_2}(1,2) = \mathbb F_2 \langle s_0, s_1, t_0, t_1 \mid t_i s_j = \delta_{ij},\; s_0 t_0 + s_1 t_1 = 1 \rangle .$$

The relations are the algebraic shadow of the Cuntz algebra $\mathcal O_2$. They say $R \cong R^2$ as right $R$-modules: the ring is isomorphic to two copies of itself. Note $t_0 s_0 = 1$ while $s_0 t_0$ is a proper idempotent, so the ring is one-sided in places. Write $R^\times$ for the group of invertible elements. Since $R$ is finitely generated over a finite field, $R^\times$ is countable.

Theorem 1.1 (Chapter 3).

$$L_{\mathbb F_2}(1,2)^\times \ \text{ is not sofic.}$$

The proof works inside a finitely generated subgroup, which suffices because soficity passes to subgroups. A complete prefix code is a finite set of pairwise prefix-incomparable binary words whose cylinders partition all infinite binary strings; once its $n$ words are ordered, it induces a unital ring isomorphism

$$\Theta_E : M_n(R) \xrightarrow{\ \sim\ } R .$$

Applying this to a specific nine-word code $D$ realizes the elementary matrix group $\mathrm{EL}_9(R)$ inside $R^\times$ as $G := \mathrm{EL}_D(R)$. By the Ershov–Jaikin-Zapirain theorem, elementary groups over a finitely generated ring have Kazhdan’s property $(T)$ once $n \ge 3$, so $G$ is a property-$(T)$ group.

Property $(T)$ is a spectral-gap condition: every unitary representation with almost invariant unit vectors has a genuine nonzero invariant vector. Its relevance here is geometric rather than analytic. Given permutations modeling a finite generating set, the generator graph joins each point of $Y_n$ to its image under each generator. Kun’s theorem says the generator graph of a sofic approximation to a property-$(T)$ group becomes a disjoint union of uniformly expanding graphs after changing $o(|Y_n|)$ edges,⁴ where a component $C$ is a $\gamma$-expander when

$$\lvert \partial_C U \rvert \ \ge\ \gamma \min\{\lvert U \rvert,\, \lvert C \setminus U \rvert\} \qquad (U \subseteq C).$$

The Kun–Thom centralizer theorem then says: if $K$ has property $(T)$, $J$ is finitely generated, and a sofic approximation of $K \times J$ has a single uniformly expanding $K$-generator graph on the whole of $Y_n$, then $J$ is LEF.

The gap between the two is the whole difficulty. Kun’s decomposition delivers many expanding components; Kun–Thom needs one. The gap is real, not technical: $\Lambda = \mathrm{SL}_3(\mathbb Z)$ has property $(T)$ and is residually finite, $B = \mathrm{BS}(2,3)$ is sofic but not residually finite and hence not LEF, and $\Lambda \times B$ is nonetheless sofic. Expanding $\Lambda$-components alone cannot force a commuting factor to be LEF.

The chapter’s bridge is a group-theoretic criterion. Suppose $\Gamma \le G$ are infinite finitely generated property-$(T)$ groups with

$$G = \langle \Gamma, t_1, \ldots, t_m \rangle, \qquad t_i \Gamma t_i^{-1} \le \Gamma \quad (1 \le i \le m),$$

and a finitely generated $J \le G$ satisfies

$$[\Gamma, J] = 1, \qquad \Gamma \cap J = \{1\}, \qquad t_1 J t_1^{-1} \le \Gamma .$$

Then soficity of $G$ implies $J$ is LEF. The extra nesting, in which conjugation by $t_1$ pushes both commuting factors inside $\Gamma$, is exactly what the $\mathrm{SL}_3(\mathbb Z) \times \mathrm{BS}(2,3)$ example lacks.

The argument applies Kun’s theorem separately to the $\Gamma$- and $G$-generator graphs, giving two different partitions of $Y_n$. A median-normalized size function

$$f(z) = \frac{M(z)}{M(z) + m_A}, \qquad M(z) = \lvert C(z) \rvert$$

on each $G$-component $A$ is almost nondecreasing along the permutations approximating the $t_i$ and almost preserved by $\Gamma$-generators. Because permutations conserve total increase and decrease, expansion pins $f$ near $1/2$ outside a negligible set, which forces each transported $\Gamma$-component to have roughly the same size as its target, so it fills more than half of it and the matching is injective. That injectivity lets the argument descend to a single expanding component and invoke Kun–Thom.

Finally the Leavitt configuration supplies the pieces. Inside $G = \mathrm{EL}_D(R)$ the chapter builds a property-$(T)$ subgroup $\Gamma$ acting inside the cylinder $[0]$, units $u, v$, and a subgroup $J \cong V$, Thompson’s group $V$, acting inside the disjoint cylinder $[1000]$, with conjugation by $u$ moving $J$ into $[0001] \subseteq [0]$. Thompson’s group $V$ is finitely presented, infinite and simple, and no such group is LEF. So $G$ cannot be sofic, and neither can $R^\times$.

What the counterexample does and does not break

The theorem removes an assumption; it does not by itself falsify the conjectures that relied on it.

The surjunctivity entry is the sharpest. Surjunctivity passes to subgroups, so if $R^\times$ is surjunctive, it is the first surjunctive nonsofic group; if it is not, Gottschalk’s conjecture is false. Either way the next result comes from the same object.

Group von Neumann algebras

Chapter 4 moves to operator algebras. For a countable discrete group $G$, the left regular representation is

$$\lambda_G : G \to \mathcal U(\ell^2(G)), \qquad \lambda_G(g)\delta_h = \delta_{gh},$$

and the group von Neumann algebra is the double commutant

$$L(G) = \lambda_G(G)'' , \qquad \tau_G(x) = \langle x \delta_1, \delta_1 \rangle .$$

Concretely, $L(G)$ is the weak closure of finite complex combinations $\sum_g a_g \lambda_G(g)$, with canonical trace $\tau_G(x) = a_1$. It is an analytic completion of the group algebra, and completion loses information: it is a coarse invariant by construction.

A group is ICC if it is infinite and every nonidentity conjugacy class is infinite. For infinite $G$, the classical criterion of Murray and von Neumann says $L(G)$ is a $\mathrm{II}_1$ factor, an infinite-dimensional factor with a faithful normal tracial state, precisely when $G$ is ICC.

How coarse is the invariant? Connes’s classification of injective factors gives the extreme case: every amenable ICC group has the same group factor, the hyperfinite $\mathrm{II}_1$ factor. All amenability collapses to one point.

Property $(T)$ is the opposite of amenability, and was expected to prevent the collapse entirely.

Conjecture (Connes). Let $G$ and $H$ be countable ICC groups with property $(T)$. If $L(G) \cong L(H)$, then $G \cong H$.

It appears as Problem 1 in the appendix to Chapter 5 of his 1994 monograph.⁵

The surrounding notion is W*-superrigidity: a countable group $G$ is W*-superrigid if $L(G) \cong L(H)$ implies $G \cong H$ for every countable $H$, so the group is fully recoverable from its factor. Such groups exist; the first examples came from Ioana, Popa and Vaes,⁶ and the first with property $(T)$ from Chifan, Ioana, Osin and Sun. The question was whether property $(T)$ alone forces it.

The fiber is infinite

Theorem (Chapter 4). There exist finitely generated ICC property-$(T)$ groups $\Lambda, \Gamma_0, \Gamma_1, \Gamma_2, \ldots$ that are pairwise nonisomorphic and satisfy

$$L(\Gamma_n) \cong L(\Lambda) \qquad (n \ge 0),$$

with $\Gamma_n$ containing a copy of $\Gamma_0$ of index $2^{4n}$, so the family is mutually commensurable.

The pair $\Lambda, \Gamma_0$ already refutes Connes’s conjecture. The infinite family does more. In his Madrid ICM address, Popa proved the group-factor functor is at most countable-to-one on ICC property-$(T)$ groups,⁷ and later asked whether the fibers are in fact finite. They are not, and his countability bound is sharp.

ExpectedOne group per factor

Property (T ) is a rigidity property: it forbids almost-invariant vectors, and Connes's classification had shown that dropping it collapses every amenable ICC group onto one hyperfinite factor. Rigidity was expected to invert the collapse.

ActualA countably infinite fiber

All of Λ, Γ₀, Γ₁, … are finitely generated ICC property-(T ) groups, pairwise nonisomorphic, pairwise commensurable, with one common group von Neumann algebra. Γ₀ sits inside Γ_n with index 2⁴ⁿ.

Connes's rigidity conjecture

false; the pair Λ, Γ₀ already suffices

Popa: are the fibers finite?

no; this fiber is countably infinite

Popa: are the fibers countable?

yes, and the bound is now sharp

W*-superrigid property-(T ) groups

still exist; property (T ) is not sufficient

The mechanism is a deliberate leak in the invariant. For a countable abelian $K$-module $A$, Fourier transform gives

$$L(A \rtimes K) \cong L^\infty(\widehat A, m_{\widehat A}) \rtimes K .$$

The right-hand side depends on the Haar probability space of the dual $\widehat A$ and on the $K$-action, but not on the compact group law of $\widehat A$. Two different group structures on the same measured $K$-space give the same crossed product, and hence the same factor, while their discrete duals can be nonisomorphic groups.

The elementary model is the four-point space $\mathbb F_2^2$. Coordinatewise addition makes it the Klein four-group. The binary carry

$$(x, y) \boxplus (x', y') = (x + x',\, y + y' + xx')$$

makes the same four points into $\mathbb Z/4\mathbb Z$. Same set, same uniform Haar measure, different group.

The construction globalizes this. The acting group is a torsion-free ICC property-$(T)$ group

$$K = \ker\Bigl(\mathrm{SL}_4(\mathbb Z[t]) \xrightarrow{\ g(t) \mapsto g(0) \bmod 3\ } \mathrm{SL}_4(\mathbb F_3)\Bigr),$$

which surjects onto $\mathrm{SL}_4(\mathbb F_2[t])$. With $V = \mathbb F_2[t]^4$ and the divided-square module

$$B = \operatorname{span}_{\mathbb F_2}\{ v \otimes v : v \in V \}, \qquad D = V \oplus B,$$

the dual $\widehat D$ gives the common compact $K$-space. Carrying the binary rule to a compact group $C_0$ with the same measured $K$-action and setting

$$\Lambda = D \rtimes K, \qquad \Gamma_0 = \widehat{C_0} \rtimes K,$$

Fourier transform identifies $L(\Gamma_0) \cong L(\Lambda)$, while order-four torsion distinguishes the two groups as abstract groups.

Verifying that $\Lambda$ is ICC and has property $(T)$ suffices for the pair, because both properties transfer across an isomorphism of group factors: by the Murray–von Neumann criterion the ICC property is detected by factoriality, and by the Connes–Jones characterization an ICC group has property $(T)$ if and only if its group factor does.

Shifting the carry by $n$ coefficient positions in each of the four directions produces compact groups $C_n$ with the same measured $K$-action, hence $\Gamma_n = \widehat{C_n} \rtimes K$ with $L(\Gamma_n) \cong L(\Lambda)$. Pontryagin duality embeds $\Gamma_0$ in $\Gamma_n$ with index $2^{4n}$, and the finite-orbit part of the intrinsic quotient $E_n[2]/2E_n$ has order $2^{4n}$, an invariant computed from the abstract group that recovers $n$, which is what proves the family pairwise nonisomorphic.

Infinite fibers were already known outside the property-$(T)$ world: for a nontrivial finite abelian $H_0$ and $n \ge 3$, infinitely many pairwise nonisomorphic $H$ satisfy $L(H) \cong L(H_0 \wr \mathrm{PSL}_n(\mathbb Z))$, but that wreath product lacks property $(T)$. The new content is an infinite fiber sitting entirely inside the class where the countability bound applies.

Reading the two together

Both results are negative, and both are negative in the same way: a property that looked like it should transmit rigidity turns out not to.

Property $(T)$ appears in both chapters, in opposite roles. In Chapter 3 it is the tool: it forces sofic approximations to be expanders, and expansion is what lets the argument match components and isolate a single one. In Chapter 4 it is the hypothesis that fails to deliver: spectral rigidity of the group does not survive the passage to the weak closure, because the crossed product only remembers a measure space and an action.

Neither proof needs a stability hypothesis or a computational reduction. Both are ordinary mathematics, checkable line by line, and OpenAI published Lean 4 certificates alongside the paper.⁸ Both close questions that had resisted the standard machinery for decades. That the arguments were produced by a model rather than a person is a fact about provenance; the theorems stand on their statements.

References

Footnotes

1. OpenAI, “Ten Advances in Mathematics and Theoretical Computer Science,” 2026. https://openai.com/index/ten-advances-in-mathematics/ ↩

2. Misha Gromov, “Endomorphisms of symbolic algebraic varieties,” Journal of the European Mathematical Society, 1999. https://doi.org/10.1007/PL00011162 ↩

3. Vladimir Pestov, “Hyperlinear and Sofic Groups: A Brief Guide,” Bulletin of Symbolic Logic, 2008. https://arxiv.org/abs/0804.3968 ↩

4. Gábor Kun, “On sofic approximations of property (T) groups,” arXiv, 2019. https://arxiv.org/abs/1606.04471 ↩

5. Alain Connes, “Noncommutative Geometry,” Academic Press, 1994. https://alainconnes.org/wp-content/uploads/book94bigpdf.pdf ↩

6. Adrian Ioana, Sorin Popa, and Stefaan Vaes, “A class of superrigid group von Neumann algebras,” Annals of Mathematics, 2013. https://arxiv.org/abs/1007.1412 ↩

7. Sorin Popa, “Deformation and rigidity for group actions and von Neumann algebras,” ICM Madrid, 2006. https://arxiv.org/abs/math/0603038 ↩

8. OpenAI, “ten-proofs: Lean certificates accompanying proofs in mathematics and theoretical computer science,” GitHub, 2026. https://github.com/openai/ten-proofs ↩