Boone-Higman embeddings of $\mathrm{Aut}(F_n)$ and mapping class groups of punctured surfaces

TL;DR

The paper proves that Aut$(F_n)$ embeds into a finitely presented simple group, establishing the Boone–Higman conjecture in a permutational form via twisted Brin–Thompson groups. The authors analyze the action of Aut$_G(G*F_n)$ on Hom$_G(G*F_n,G)$ and show this action is of type (A) for suitable G, yielding a finite presentability for the resulting simple groups and, in turn, PBH for Aut$(F_n)$. They further show PBH is equivalent to embeddings into finitely presented simple highly transitive (or MIF) groups, making twisted Brin–Thompson groups universal for this class; this leads to PBH for a wide array of related groups, including mapping class groups of punctured surfaces, braid groups, loop/ribbon braid groups, and several Artin groups. The results provide a robust permanence framework (e.g., under free products) and illuminate the relationship between BH and PBH, while also delivering concrete PBH conclusions for many geometrically and algebraically significant groups.

Abstract

We prove that the groups $\mathrm{Aut}(F_n)$ satisfy the Boone-Higman conjecture for all $n$, meaning each $\mathrm{Aut}(F_n)$ embeds in a finitely presented simple group. In fact, we prove that each $\mathrm{Aut}(F_n)$ 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 $\mathrm{Aut}(F_n)$ also confirms the conjecture for all groups (virtually) embedding into some $\mathrm{Aut}(F_n)$, 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.

Theorems & Definitions (28)

• Theorem A
• Corollary B
• Theorem C
• Corollary D
• Theorem E
• Corollary F
• Proposition A
• proof
• Proposition B
• proof