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Soficity Conjecture

The soficity conjecture is the affirmative answer to the question of whether every countable group is a sofic group (Gromov 1999, Weiss 2000). The question has a negative answer (OpenAI 2026, Fournier-Facio 2026, Kun and Thom 2026).

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.

The proof more specifically constructs a finitely generated nonsofic subgroup of L_(F_2)(1,2)^×. Kun and Thom (2026) give a strengthening of the centralizer argument that also applies to this example, while Fournier-Facio (2026) constructs a finitely presented torsion-free nonsofic group.

Sandhu and Thom (2026) announced a torsion-free, two-dimensional hyperbolic group that is not a sofic group. It is realized as the fundamental group of a finite triangle complex. The construction originated in an AI-generated argument and was subsequently reworked by Thom. It uses ingredients from an OpenAI construction of a non-residually finite hyperbolic group. Independent verification of the complete argument has not been established here.

See also

Countable Set, Group, Sofic Group

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References

Fournier-Facio, F. "A Torsion-Free Non-Sofic Group." 14 Aug 2026. https://arxiv.org/abs/2608.02025v2.Gromov, M. "Endomorphisms of Symbolic Algebraic Varieties." J. Eur. Math. Soc. 1, 109-197, 1999. https://doi.org/10.1007/PL00011162.Kun, G. and Thom, A. "Nonsofic Wreath Products of Residually Finite Groups." 20 Aug 2026. https://arxiv.org/abs/2608.06222v3.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. "Nonsofic Groups Exist." Ch. 3 in Ten Advances in Mathematics and Theoretical Computer Science. Aug. 1, 2026 (revised Aug. 6, 2026). https://cdn.openai.com/pdf/ten-proofs-oai.pdf.Sandhu, H. and Thom, A. "An Example of a Non-Sofic Hyperbolic Group." 17 Oct 2026. https://arxiv.org/abs/2610.24836.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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