The Boone-Higman conjecture: groups with solvable word problem embed in finitely presented simple groups

Theorem 1.1: a finitely generated group has decidable word problem if and only if it embeds in a finitely presented simple group. The route is Theorem 1.2: every such group embeds in a finitely presented group with a faithful two-transitive action on a countable set with finitely generated point stabilizers, built as a mapping torus of an affine Steinberg group over finitely presented $\mathbb F_2$-algebras, which Zaremsky's criterion for twisted Brin-Thompson groups turns into a finitely presented simple overgroup. The higher-finiteness strengthening is the companion's separate entry; no explicit bound on the size of the presentation is given.

Theorem 1.1: a finitely generated group has decidable word problem if and only if it embeds in a finitely presented simple group. The route is Theorem 1.2: every such group embeds in a finitely presented group with a faithful two-transitive action on a countable set with finitely generated point stabilizers, built as a mapping torus of an affine Steinberg group over finitely presented $\mathbb F_2$-algebras, which Zaremsky's criterion for twisted Brin-Thompson groups turns into a finitely presented simple overgroup. The higher-finiteness strengthening is the companion's separate entry; no explicit bound on the size of the presentation is given.