We study the coarse geometry of the mapping class group of a compact orientable surface. We show that, apart from a few low-complexity cases, any quasi-isometric embedding of a mapping class group itself agrees up to bounded distance with a left multiplication. In particular, such a map is a quasi-isometry. This is a strengthening of the result of Hamenst\"adt and of Behstock, Kleiner, Minsky and Mosher that the mapping class groups are quasi-isometrically rigid. In the course of also develop the general theory of coarse median spaces and median metric spaces with a view to other appications elsewhere.
Preprint, Warwick, March 2015.
Revised, March 2016.
Pacific J. Math. 293 (2018) 1-73.