Embedding groups into boundedly acyclic groups

We show that the 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 Fₙ embeds quasi-isometrically into a boundedly acyclic group of type Fₙ that has no proper finite index subgroups. This improves a result of Bridson {Br98} and a theorem of Fournier-Facio--L\"oh--Moraschini [Theorem 2]{FFCM21}. Second, every group of type Fₙ embeds quasi-isometrically into a $5$-uniformly perfect group of type Fₙ. 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.