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The permutational Boone–Higman conjecture

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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 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.

  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).

References

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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