Embedding groups into boundedly acyclic groups

Abstract 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 embeds quasi‐isometrically into a boundedly acyclic group of type 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 embeds quasi‐isometrically into a 5‐uniformly perfect group of type . 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 ‐labeled Thompson group and .