Embedding groups into boundedly acyclic groups

TL;DR

This paper proves that certain Thompson and Brin--Thompson groups are boundedly acyclic, enabling new embedding results for groups of type F_n into boundedly acyclic and simple groups, advancing understanding of group embeddings.

Contribution

It establishes bounded acyclicity for s{ extbackslash phi}-labeled Thompson groups and uses this to derive novel embedding theorems for groups of type F_n and simple groups.

Findings

s{ extbackslash phi}-labeled Thompson groups are boundedly acyclic.

Every group of type F_n embeds into a boundedly acyclic group of the same type.

Every finitely generated group embeds into a finitely generated boundedly acyclic simple group.

Abstract

We show that the \s{\phi}-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--L\"oh--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 $\phi$-labeled Thompson group $V_\phi(G)$ and $F_\phi(G)$.