Quasi-isometry

Summary

A quasi-isometry is a map between metric spaces that distorts distances by at most a fixed factor and a fixed additive error. In addition, its image comes within a bounded distance of every point. It ignores all small-scale structure and keeps only the large-scale shape of a space. Two spaces joined by a quasi-isometry are called quasi-isometric, and this is an equivalence relation. It is the basic notion of sameness in geometric group theory, since all word metrics of a finitely generated group are quasi-isometric to one another.