Existence of non-sofic groups
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Does every group admit arbitrarily good finite-permutation approximations, that is, is every group sofic? The focused target is an explicit finitely presented nonsofic group.
Why it matters and what counts as progress
Why it matters: It asks whether a general finite-approximation principle holds throughout group theory, even for finitely presented groups.
What counts as progress: A group with a proof excluding all sofic approximations, or a soficity theorem covering a previously unresolved class.
How the problem developed
Mikhail Gromov Introduced sofic approximations in the setting that led to the question whether every countable group is sofic.
- 1 Aug 2026 · Reported result [3]
OpenAI Reported a finitely generated non-sofic subgroup of the unit group of the binary Leavitt algebra, providing an explicit example and a Lean formalization.
AI contribution
Ten Advances in Mathematics and Theoretical Computer Science
OpenAI gives a finitely presented group that lacks finite permutation approximations, establishing the existence of non-sofic groups. Its construction combines a Kun–Thom criterion with an expander-matching argument involving binary Leavitt algebra and Thompson’s group V.
Sources
- Mikhail Gromov, Endomorphisms of symbolic algebraic varieties
- Benjamin Weiss, Sofic groups and dynamical systems
- OpenAI, Ten advances in mathematics and theoretical computer science
- OpenAI, Ten advances: research account
- OpenAI ten-proofs formalization manifest
- Lean proof of a non-sofic group
- Sienicki et al., chapter-by-chapter mathematical review
- Fournier-Facio, later non-sofic construction
- Kun–Thom, later use of the non-sofic criterion
Website timeline
Publication dates and editorial changes.