Embed groups into bounded acyclic groups

We first discuss various embedding results for groups in the literature. Then we talk about how could one embed a group of type $F_n$ into a group of type $F_n$ with no proper finite index subgroup quasi-isometrically. The embedding we have uses the so called labelled Thompson groups, and it is functorial. We also show that the labeled Thompson group is always bounded acyclic. As a corollary, one could embed any group of type $F_n$ into a bounded acyclic group of type $F_n$ quasi-isometrically. This is based on a joint work with Fan Wu, Mengfei Zhao and Zixiang Zhou.