Remark: Family 250. The Boone–Higman conjecture and 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 $F_{\infty }$: a classifying space with finitely many cells in each dimension. A single group of type $F_{\infty }$ can also contain every finitely presented group.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.