Boone–Higman Conjecture

The Boone–Higman conjecture asks whether every finitely generated group with solvable word problem embeds into a finitely presented simple group. It grew out of the 1974 theorem of Boone and Higman characterizing solvability of the word problem algebraically: a finitely generated group has solvable word problem if and only if it embeds into a simple subgroup of some finitely presented group [BooneHigman1974]. Since finitely presented simple groups have solvable word problem, the conjectured embedding would give a clean equivalence between the computability condition and a purely algebraic one.

The original theorem thus provides the weaker embedding, with the simple group sitting inside, rather than equal to, a finitely presented group. Recent years have seen rapid progress, surveyed by Belk, Bleak, Matucci, and Zaremsky [BelkEtAl2025Survey]: the full conclusion is now established for many major classes, notably including all hyperbolic groups [BelkEtAl2026Hyperbolic].

The general case is open. A resolution requires either an embedding construction valid for every finitely generated group with solvable word problem, or a group with solvable word problem that admits no embedding into any finitely presented simple group.