Free Group Factor Problem: OpenAI Claims L(F2) = L(F3)

About this video

The free group on two letters, F2, is every word you can write with a, b and their inverses without anything cancelling; F3 does the same with three letters. Each group builds an infinite-dimensional operator algebra, its group von Neumann algebra L(Fk). Whether L(F2) and L(F3) are the same algebra is one of the best-known open problems in operator algebras. Work of Dykema and Radulescu showed an all-or-nothing answer: either every free group factor is isomorphic to every other, or no two are. OpenAI claims a proof that L(F2) and L(F3) are isomorphic, so all interpolated free group factors, including L(F-infinity), are the same algebra, with fundamental group all positive real numbers. This is OpenAI's preprint, not yet peer reviewed. The animation draws the Cayley trees of F2 and F3 to word length 4 with the computed word counts (4, 12, 36, 108 against 6, 30, 150, 750), and lights up the even-length words of F2, a subgroup of index 2 that is itself a free group on three letters (Schreier's formula; checked to reach all 1,093 even words up to length 6). The trees merging at the end is an illustration only: the trees are not the algebras.

Papers (OpenAI, catalogue result 287):
- An isomorphism of the free group factors: github.com/openai/math/blob/main/preprints/An-isomorphism-of-…

The animation illustrates the result; it is not a proof.

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Every result explained: gradientdamage.com/p/287