Boone–Higman embeddings with higher finiteness

2 · Going further

For a finitely generated group $G$, the word problem asks whether a word in its generators and their inverses represents the identity. The main manuscript shows the equivalence: $G$ has decidable word problem if and only if $G$ embeds in a finitely presented simple group. An embedding preserves the group operations and identifies no distinct elements; simplicity means having no normal subgroups other than the identity subgroup and the whole group. The manuscript presents this equivalence as the resolution of the Boone–Higman conjecture.

The second manuscript strengthens the forward embedding statement: every finitely generated group with decidable word problem embeds in a simple group of type $F_{\infty }$. Here type $F_{\infty }$ means admitting a classifying space with finitely many cells in each dimension. The source describes this as resolving the higher-finiteness strengthening of the Boone–Higman conjecture. The abstracts state the embedding results but do not describe the construction or steps of the argument, so no more specific proof strategy can be extracted from them.

The third manuscript supplies a different, universal construction: a single group $U$ of type $F_{\infty }$ containing every finitely presented group. It also characterizes the finitely generated subgroups of $U$, up to isomorphism, as exactly the finitely generated recursively presented groups. The source says this answers the $F_{\infty }$ form of the higher-dimensional Higman embedding question and marks this manuscript as Lean-formalized. Unlike the second manuscript's targets, this universal group is not stated to be simple.

Useful notions

• Groups, generators, and relations
• The word problem and decidability
• Embeddings and subgroups
• Normal subgroups and simplicity
• Classifying spaces and cells

What was known before / what is new

Limits and assumptions

The Boone–Higman equivalence is stated for finitely generated groups, and the second manuscript's embedding result requires decidable word problem. The universal containment statement in the third concerns all finitely presented groups; its broader subgroup characterization concerns finitely generated recursively presented groups. Type $F_{\infty }$ requires finitely many cells in each dimension, not finitely many cells altogether. The supplied material marks the third manuscript as Lean-formalized, but gives no finer breakdown of formalization coverage and does not establish formal verification of the first two manuscripts. The abstracts do not provide argument details or identify further questions that remain open.

Mini glossary

Decidable word problem

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The existence of an algorithm that always determines whether a word in group generators and their inverses represents the identity.

Embedding

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An injective map between groups that preserves their operations.

Finitely presented group

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A group describable using finitely many generators and finitely many defining relations.

Simple group

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A group whose only normal subgroups are the identity subgroup and the whole group.

Type F∞

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The property of admitting a classifying space with finitely many cells in each dimension.

Recursively presented group

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A group admitting a presentation whose defining relations can be enumerated by an algorithm.

Keywords : Boone–Higman conjecture · Word problem · Type F∞

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Manuscripts in this family