Embedding groups into boundedly acyclic groups

2024/07/10 by Fan Wu, Xiaolei Wu, Wu, Fan +4 · 3 citations
Mathematics · #21J06 #57M07 #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #K-Theory and Homology (math.KT)

paper · pdf · doi:10.48550/arxiv.2407.07703

openalex publication_date 2024/07/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that the \sϕ-labeled Thompson groups and the twisted Brin--Thompson groups are boundedly acyclic. This allows us to prove several new embedding results for groups. First, every group of type F_n embeds quasi-isometrically into a boundedly acyclic group of type F_n that has no proper finite index subgroups. This improves a result of Bridson and a theorem of Fournier-Facio--Löh--Moraschini. Second, every group of type F_n embeds quasi-isometrically into a 5-uniformly perfect group of type F_n. Third, using Belk--Zaremsky's construction of twisted Brin--Thompson groups, we show that every finitely generated group embeds quasi-isometrically into a finitely generated boundedly acyclic simple group. We also partially answer some questions of Brothier and Tanushevski regarding the finiteness property of ϕ-labeled Thompson group V_ϕ(G) and F_ϕ(G).

Cited by

• Stability, approximable quotients, and higher property (T)
• Embedding groups into acyclic groups
• Infinitely presented simple groups separated by homological finiteness properties

Related

• Embedding groups into acyclic groups
• A taste of twisted Brin-Thompson groups
• Quasi-isometrically embedded subgroups of Thompson's group F
• Finite presentability of twisted Brin-Thompson groups
• Embedding Right-Angled Artin Groups into Brin-Thompson Groups