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Boone–Higman embeddings with higher finiteness
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.
Manuscripts in this family · 3
Finite algebraic envelopes and the Boone–Higman conjecture
Sep 23, 2026·paper.pdf
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.
Simple F∞ overgroups of groups with decidable word problem
Sep 23, 2026·paper.pdf
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.
A universal group of type F∞
Sep 23, 2026·paper.pdfLean
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.