OpenAI research collection · 3 manuscripts · Source checked 2026-10-19
A finitely generated group has decidable word problem exactly when it embeds in a finitely presented simple group, proving the Boone–Higman conjecture. The target can have type F_(∞): a classifying space with finitely many cells in each dimension. A single group of type F_(∞) can also contain every finitely presented group.
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We prove the Boone–Higman conjecture. A finitely generated group has decidable word problem if and only if it embeds in a finitely presented simple group.
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Every finitely generated group with decidable word problem embeds in a simple group of type F_(∞). This resolves the higher-finiteness strengthening of the Boone–Higman conjecture.
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We construct a single group of type F_(∞) containing every finitely presented group. Its finitely generated subgroups, up to isomorphism, are exactly the finitely generated recursively presented groups. This answers the F_(∞) form of the higher-dimensional Higman embedding question.
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New mathematical reasoning could help AI tackle harder scientific and computational problems. How reliably these skills transfer beyond mathematics will determine how much they contribute to general intelligence.
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