论置换圈积的自相似性及其在有限表现单群中的嵌入
On the Self-Similarity of Permutational Wreath Products and Their Embedding into Finitely Presented Simple Groups
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中文总结 AI 辅助
该研究探讨置换圈积的自相似性，证明特定条件下Scott–Röver–Nekrashevych群的有限表现性与几乎单性，及\boldsymbol{\boldsymbol{Z}^d \rtimes_X G}嵌入有限表现单群，为Boone–Higman猜想提供新群族。
中文摘要 AI 辅助
本工作研究形如\boldsymbol{A \rtimes_X G}的置换圈积的自相似性，其中\boldsymbol{A}是有限生成阿贝尔群，\boldsymbol{G}是自相似群（置换圈积\boldsymbol{A \rtimes_X G}也称为灯夫群）。当\boldsymbol{G}是非挠收缩群时，我们证明在一定条件下，Scott–Röver–Nekrashevych群\boldsymbol{V_m(\boldsymbol{Z}^d \rtimes_X G)}是有限表现且几乎单的；此外，我们证明\boldsymbol{\boldsymbol{Z}^d \rtimes_X G}可嵌入有限表现单群，这为满足Boone–Higman猜想的群提供了新的族。
英文摘要
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.
发表机构
- Universidade de Brasília（巴西利亚大学）
- Universidade Estadual do Norte do Paraná（北帕拉那州立大学）
机构由 AI 辅助整理，请以论文原文为准。
