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 R & ouml;ver-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.