Non-sofic groups

On 1 August 2026, OpenAI announced the solution of 10 major open problems, including the first construction of a non-sofic group. The proof crucially relies on work of Kun and Kun--Thom to prove the technical Proposition 2.3, which shows that a certain configuration of property (T) and non-LEF groups are such that the container group cannot be sofic. This is then applied to a construction involving elementary matrices over the binary Leavitt algebra - which have (T) by work of Ershov--Jaikin-Zapirain - and Thompson's group V - which is not LEF.

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Below is an alternative construction of groups in this configuration, which avoids Leavitt algebras and Thompson groups, and gives torsion-free examples. The introduction also contains some comments about the solution.

A torsion-free non-sofic group, 4 pages.

The first version was added to this homepage in the evening of 1 August 2026, but I made small corrections since then. In particular, I was critical of the phrasing in the first announcement, which talked of "no progress for at least a decade" on any of the 10 problems; this wording has since been redacted. To me this shows that Proposition 2.3 is the key to this solution. If you want to learn more about its proof, see Andreas Thom's beautiful explanation on MathOverflow.

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Videos:

• A video for the channel of Álvaro Lozano-Robledo where I talk about it starting from what a group is.

• If you speak German (or if the youtube translation is any good), an interview of Andreas Thom with DorFuchs.

• A meeting at Weizmann, with an introduction to sofic groups by Benjy Weiss, and an explanation of the proof by Alon Dogon.