Isomorphism of the free group factors

Proves, with a Lean check, that the free group factors L(F₂) and L(F₃) are isomorphic, settling a decades-old problem.

1 paper September 23, 2026 Explained by AI

A free group is made of words in some letters and their inverses, where a letter cancels next to its inverse. Each group gives an operator algebra. For decades it was open whether free groups on two and three letters give the same algebra. This paper constructs an isomorphism between them, so the answer is yes. Lean checks that all interpolated free group factors are isomorphic.

First step

1943

Francis Murray and John von Neumann, introduction of the free group factors

Significance

Landmark

The free group factor isomorphism problem is one of the best-known open problems in operator algebras.

Trust

Formal proof listed

Field

Operator algebras

1 paper

The explanation

Write words in the letters a and b and their inverses A and B. Cancel any letter that sits next to its inverse, so aA disappears. What is left is the free group on two letters, F₂. It has 4 words of length one, 4 × 3 = 12 of length two and 36 of length three. With three letters, F₃ has 6, 30 and 150. These are different groups; no relabelling turns one into the other.

Each group gives an operator algebra, its group von Neumann algebra L(G). Every group element acts on square-summable sequences by shifting them. The algebra is the closure of these shifts. The free group factors arose in the foundational work of Murray and von NeumannSource 1. Whether L(F_m) and L(F_n) are isomorphic for distinct m and n became a central problem in classifying finite factorsSource 2. Kadison is credited with raising the question.

Free probability gave tools but no answer. Dykema and Rădulescu built interpolated factors L(F_r) for every real r > 1. Their amplification formula says a resized copy of L(F_s) by factor t is L(F_r) with r = 1 + (s − 1)/t²Source 3. They also gave a dichotomy: either all interpolated factors are isomorphic, or no two areSource 4. So one isomorphism between two distinct ranks settles everything. The obstacle was to build one.

The paper proves that L(F_n) and L(F_(n+1)) are isomorphic for every n of at least 3Source 5. It applies this at ranks 3 and 4, giving L(F₃) ≅ L(F₄) ≅ L(F₅). Now resize both ends by √2, so t² = 2. The formula gives 1 + (3 − 1)/2 = 2 and 1 + (5 − 1)/2 = 3. So L(F₂) ≅ L(F₃)Source 6. With the dichotomy, all interpolated free group factors, including the infinite-rank one, are isomorphicSource 7. The paper also deduces that their fundamental groups are all positive realsSource 8. It answers negatively Voiculescu's question on free entropy dimensionSource 9. That question asked whether a numerical invariant of generators depends only on the algebra they generate.

The construction uses a flow, a continuous family of automorphisms built from a differential equation. Each step moves the n generators very little but makes a word in them approximate an extra generator. Repeating this absorbs the extra generator in the limit.

How sure is this? A computer proof checker called Lean has a formal proof. It covers the isomorphism of interpolated free group factors for all parameters r, s > 1, including infinity. The fundamental-group statement is a further consequence and is not a separate Lean statement. No specialist review of the paper has been published.

What it doesn't show. This does not say the free groups themselves are isomorphic; F₂ and F₃ remain different groups.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Landmark: it resolves the free group factor problem. Lean formally checks the isomorphism of all interpolated free group factors; no specialist review is published yet.

Words used

Free group F_n: All words in n letters and their inverses, where the only rule is cancelling a letter next to its inverse.
Group von Neumann algebra L(G): The operator algebra generated by the shifts that group elements make on square-summable sequences.
Factor: A von Neumann algebra whose only central elements are multiples of the identity.
Isomorphism: A one-to-one match between two algebras that keeps all the operations; here it also keeps the trace.
Interpolated free group factor: A factor L(F_r) defined for every real r > 1, agreeing with L(F_n) at whole numbers.
Amplification: Resizing a factor by a factor t, by taking a corner or a larger matrix version.
Fundamental group of a factor: The set of scales t at which the factor is isomorphic to its own resized copy.
Free entropy dimension: A number attached to a set of generators, from Voiculescu's free probability theory.

The original paper

An isomorphism of the free group factors (PDF, opens in a new tab)
September 23, 2026

Lean notes Model's reasoning summary

All quoted lines (9)

The free group factors arose in Murray and von Neumann's foundational work.
An isomorphism of the free group factors Back to text
The question whether $L(\F_m)$ and $L(\F_n)$ are isomorphic for distinct $m,n\geq2$ became a central problem in the classification of finite factors.
An isomorphism of the free group factors Back to text
The established amplification formula is $N_s^t\cong N_{1+(s-1)t^{-2}} \qquad(s>1,\ t>0);$ see.
An isomorphism of the free group factors Back to text
They also give a dichotomy: the interpolated free group factors are either all isomorphic or pairwise nonisomorphic.
An isomorphism of the free group factors Back to text
For every integer $n\geq3$ there is a unital trace-preserving normal $*$ -isomorphism $L(\F_n)\cong L(\F_{n+1}).$
An isomorphism of the free group factors Back to text
There exists a unital normal trace-preserving $*$ -isomorphism $\Phi:L(\F_2)\longrightarrow L(\F_3).$
An isomorphism of the free group factors Back to text
Together with the classical dichotomy, this proves that all interpolated free group factors, including the infinite-rank factor, are isomorphic.
An isomorphism of the free group factors Back to text
Their fundamental groups are therefore $\R_{>0}$ .
An isomorphism of the free group factors Back to text
Our isomorphisms give the opposite conclusion: Corollary … gives a negative answer and proves the same failure for $\delta_0,\delta^*$ , and $\delta^\star$ .
An isomorphism of the free group factors Back to text

Other sources

openai/math, Lean scope note for family 287 The formalization proves that interpolated free group factors with any parameters $r,s>1$ , including the infinite parameter, are normally trace-preservingly isomorphic.
openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

286Rigidity and arithmetic of lattice von Neumann algebrasOperator algebras
288Kadison's similarity conjectureOperator algebras
285Counterexamples to Baum–Connes and Kadison–KaplanskyOperator algebras
289Strong Kadison–Kastler stability and its spatial boundariesOperator algebras
290Relative bicentralizers and modular spectral recoveryOperator algebras

Browse Operator algebras Open since the 1940s Lean-checked results Landmark results