Embedding finitely presented self-similar groups into finitely presented simple groups

Abstract

We prove that every finitely presented self-similar group embeds in a finitely presented simple group. This establishes that every group embedding in a finitely presented self-similar group satisfies the Boone–Higman conjecture. The simple groups in question are certain commutator subgroups of Röver–Nekrashevych groups, and the difficulty lies in the fact that even if a Röver–Nekrashevych group is finitely presented, its commutator subgroup might not be. We also discuss a general example involving matrix groups over certain rings, which in particular establishes that every finitely generated subgroup of ${GL}_n\left(Q\right)$ satisfies the Boone–Higman conjecture.