MathDBBoone--Higman conjecture
Boone--Higman conjecture
A finitely generated group is said to have solvable word problem when an algorithm determines whether each word in a finite generating alphabet represents the identity.
Boone--Higman conjecture. Every finitely generated group with solvable word problem embeds in a finitely presented simple group.
This is a classical embedding conjecture and would give an algebraic characterization of solvability of the word problem. It is known for several families of groups, but remains open in general.
Equivalent formulations 1
Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.
The Boone–Higman conjecture
A group has solvable word problem if there is an algorithm deciding whether a word in its generators represents the identity. Boone–Higman conjecture. Every finitely generated group with solvable word problem embeds in a finitely presented simple group. The converse is known, and the conjecture asks whether the two-step Boone–Higman–Thompson embedding can be replaced by a single embedding; it is solved for several important classes of groups but remains open in general.
source: James Belk, Francesco Fournier-Facio, James Hyde and Matthew C. B. Zaremsky, “Boone-Higman embeddings of Aut(F_n) and mapping class groups of punctured surfaces”, arXiv:2503.21882 (2025).
Sources & referencesView supporting material
Primary source
Francesco Fournier-Facio, Xiaolei Wu and Matthew C. B. Zaremsky, “Abstract twisted Brin–Thompson groups”, arXiv:2603.24687 (2026).
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