Coarse separation in geometric group theory

A subset S of a metric space X is said to be coarsely separating if there exists some constant D such that for any R there exist two balls of radius R entirely contained in two different connected components of the complement of the D neighborhood of S. This notion has been introduced in the 90's and used to prove some quasi-isometric rigidity results of finitely generated groups, such as lattices in Lie groups. After giving a brief overview of these applications, I will show how it can also be used to prove the quasi-isometric rigidity of some amalgamated free products and wreath products. Joint work with Anthony Genevois and Romain Tessera.