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Soficity Conjecture

The soficity conjecture asserted that every countable group is a sofic group (Gromov 1999, Weiss 2000).

Let L_(F_2)(1,2) denote the binary Leavitt algebra over the finite field with two elements, and let a superscript × denote its group of ring units. An AI-generated proof given by OpenAI (2026) established

L_(F_2)(1,2)^× is not sofic.

This disproves the soficity conjecture. The proof more specifically constructs a finitely generated nonsofic subgroup of L_(F_2)(1,2)^×.

See also

Countable Set, Finite Field, Finitely Generated, Group, Ring Unit, Sofic Group, Subgroup

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References

Gromov, M. "Endomorphisms of Symbolic Algebraic Varieties." J. Eur. Math. Soc. 1, 109-197, 1999. https://doi.org/10.1007/PL00011162.Leavitt, W. G. "The Module Type of a Ring." Trans. Amer. Math. Soc. 103, 113-130, 1962. https://doi.org/10.1090/S0002-9947-1962-0132764-X.OpenAI. "A Counterexample to the Soficity Conjecture." Ch. 3 in Ten Advances in Mathematics and Theoretical Computer Science. Aug. 1, 2026. https://cdn.openai.com/pdf/ten-proofs-oai.pdf.Weiss, B. "Sofic Groups and Dynamical Systems." Sankhyā Ser. A 62, 350-359, 2000.

Cite this as:

Weisstein, Eric W. "Soficity Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SoficityConjecture.html

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