On the Self-Similarity of Permutational Wreath Products and Their Embedding into Finitely Presented Simple Groups

In this work, we study the self-similarity of permutational wreath products of the form $A{\wr }_{X}G$, where $A$ is a finitely generated abelian group and $G$ is a self-similar group (the permutational wreath product $A{\wr }_{X}G$ is also known as a lamplighter group). In the case where $G$ is a non-torsion contracting group, we prove that, under certain conditions, the Scott–Röver–Nekrashevych group $V_m({\mathbb{Z}}^d{\wr }_{X}G)$ is finitely presented and virtually simple. Moreover, we prove that ${\mathbb{Z}}^d{\wr }_{X}G$ embeds into a finitely presented simple group. Furthermore, this provides a new family of groups that satisfy the Boone–Higman conjecture.