Boone--Higman Embeddings for Contracting Self-Similar Groups

We give a short proof that every contracting self-similar group embeds into a finitely presented simple group. In particular, any contracting self-similar group embeds into the corresponding Röver–Nekrashevych group, and this in turn embeds into one of the twisted Brin–Thompson groups introduced by the first author and Matthew Zaremsky. The proof here is a simplification of a more general argument given by the authors, Collin Bleak, and Matthew Zaremsky for contracting rational similarity groups.