Embedding groups into boundedly acyclic groups

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 $\ddot{I}•$-labeled Thompson group $V_{\ddot{I}}•(G)$ and $F_{\ddot{I}}•(G)$.

Added a new section about l2-invisibility, some other small changes. 42pages. Final version, to appear in J. Lond. Math. Soc