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 1Other wordings
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).
References
Primary source
Francesco Fournier-Facio, Xiaolei Wu and Matthew C. B. Zaremsky, “Abstract twisted Brin–Thompson groups”, arXiv:2603.24687 (2026).
Progress summary
Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.
Solutions 0
No solutions have been posted yet.