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The permutational Boone–Higman conjecture

Let a group Γ\Gamma act faithfully on a set SS. The action is of type (A) if Γ\Gamma is finitely presented, every stabilizer Stab⁡Γ(s)\operatorname{Stab}_{\Gamma}(s) is finitely generated, and the diagonal action on S×SS\times S has finitely many orbits. Permutational Boone–Higman conjecture. Every finitely generated group with solvable word problem embeds in a group admitting an action of type (A), and hence in a finitely presented simple twisted Brin–Thompson group. This strengthens the Boone–Higman conjecture by requiring the embedding to arise from an action whose associated twisted Brin–Thompson group is finitely presented and simple; the paper proves this stronger property for several families, while the general conjecture remains open.

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.

  1. Permutational Boone--Higman conjecture

    Let G↷SG\curvearrowright S be an action of a group on a set SS, and let SVGSV_G be the associated twisted Brin--Thompson group. An action is of type (A⁡2)(\operatorname{A}_2) when it satisfies the finite-presentability condition used for twisted Brin--Thompson groups.

    Permutational Boone--Higman conjecture. Every finitely generated group with solvable word problem embeds in a group admitting an action of type (A⁡2)(\operatorname{A}_2); equivalently, it embeds in a finitely presented simple twisted Brin--Thompson group.

    This is a geometric variant of the Boone--Higman conjecture. The source notes that it implies the classical conjecture and that the equivalence of the two conjectures is a major open question.

    source: Francesco Fournier-Facio, Xiaolei Wu and Matthew C. B. Zaremsky, “Abstract twisted Brin–Thompson groups”, arXiv:2603.24687 (2026).

Sources & referencesView supporting material

Primary 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).

Additional references

2 papers in this index state this conjecture (2024–2025). The statement above is taken from the most recent of them; the others are arXiv:2405.18354.

brin-thompson-groupsgroup-theorysimple-groupsthompson-groupsword-problem

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