Boone–Higman embeddings of Aut (Fn) and mapping class groups of punctured surfaces

Source abstract

Abstract We prove that the groups satisfy the Boone–Higman conjecture for all , meaning each embeds in a finitely presented simple group. In fact, we prove that each satisfies the “permutational” Boone–Higman conjecture, which means the simple group in question can be taken to be a twisted Brin–Thompson group. A far‐reaching consequence of our approach is that finitely presented twisted Brin–Thompson groups are universal among finitely presented simple groups that are highly transitive. This is evidence toward the Boone–Higman conjecture being equivalent to its permutational version. Proving the conjecture for also confirms the conjecture for all groups (virtually) embedding into some , such as mapping class groups of non‐closed surfaces, braid groups, loop braid groups, ribbon braid groups, and certain Artin groups. This answers several questions of the first and fourth authors with Bleak and Matucci. Yet another consequence of our approach is that satisfying the permutational Boone–Higman conjecture is closed under free products and regular wreath products, which answers a question of the fourth author, confirms the conjecture for free solvable groups, and reduces the conjecture for finitely generated 3‐manifold groups to the case of non‐Seifert fibered closed graph manifolds.