On the Self-Similarity of Permutational Wreath Products and Their Embedding into Finitely Presented Simple Groups
Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Institution: Universidade de Brasília
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7207629898