Boone--Higman Embeddings for Contracting Self-Similar Groups

TL;DR

This paper presents a simplified proof that all contracting self-similar groups can be embedded into finitely presented simple groups, connecting various classes of groups through a series of embeddings.

Contribution

It provides a new, simplified proof of embeddings for contracting self-similar groups into finitely presented simple groups, extending previous results.

Findings

Every contracting self-similar group embeds into a finitely presented simple group

Embedding into R"over--Nekrashevych groups and twisted Brin--Thompson groups

Simplification of previous complex proofs

Abstract

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\"over--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.