Embedding groups into boundedly acyclic groups

We show that the $\varphi$-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 $\varphi$-labeled Thompson group $V_{\varphi }\left(G\right)$ and $F_{\varphi }\left(G\right)$.