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

  • Mailton Rego Almeida
  • Alex Carrazedo Dantas
  • Altair Santos de Oliveira-Tosti
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Abstract

In this work, we study the self-similarity of permutational wreath products of the form A wrX G, where A is a finitely generated abelian group and G is a self-similar group (the permutational wreath product A wrX 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 Vm(ℤd wrX G) is finitely presented and virtually simple. Moreover, we prove that ℤd wrX G embeds into a finitely presented simple group. Furthermore, this provides a new family of groups that satisfy the Boone--Higman conjecture.

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Cite

BibTeXRIS
BibTeX
@misc{almeida2026selfsimilarity,
  title = {{On the Self-Similarity of Permutational Wreath Products and Their Embedding into Finitely Presented Simple Groups}},
  author = {Mailton Rego Almeida and Alex Carrazedo Dantas and Altair Santos de Oliveira-Tosti},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.01868},
  eprint = {2609.01868},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.01868},
}
RIS
TY  - UNPB
TI  - On the Self-Similarity of Permutational Wreath Products and Their Embedding into Finitely Presented Simple Groups
AU  - Mailton Rego Almeida
AU  - Alex Carrazedo Dantas
AU  - Altair Santos de Oliveira-Tosti
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2609.01868
UR  - https://arxiv.org/abs/2609.01868
AB  - 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(ℤ^d wr_X G) is finitely presented and virtually simple. Moreover, we prove that 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.
ER  -