Simple overgroups of type F-infinity for groups with solvable word problem (higher-finiteness Boone-Higman)

Theorem 1.1: every finitely generated group with decidable word problem embeds in a nontrivial simple group of type $F_\infty$; with the classical converse, these are exactly the finitely generated subgroups of simple groups of type $F_\infty$. This implies the Boone-Higman conjecture. The classifying space may be infinite-dimensional; no torsion-freeness, finite cohomological dimension or type $F$ is claimed.

Theorem 1.1: every finitely generated group with decidable word problem embeds in a nontrivial simple group of type $F_\infty$; with the classical converse, these are exactly the finitely generated subgroups of simple groups of type $F_\infty$. This implies the Boone-Higman conjecture. The classifying space may be infinite-dimensional; no torsion-freeness, finite cohomological dimension or type $F$ is claimed.