Rank and Rigidity Properties of Spaces Associated to a Surface

We describe the large scale geometry of the mapping class group and of the pants graph (or equivalently the Teichmüller space in the Weil-Petersson metric) of a compact orientable surface, from the point of view of coarse median spaces. We derive various results about coarse rank and quasi-isometric rigidity of such spaces. In particular, we show that a quasi-isometric embedding of a mapping class group into itself is a bounded distance from a left multiplication, generalising the result of Hamenstädt and of Behrstock, Kleiner, Minsky and Mosher. We show that, apart from finitely many cases, the pants graphs of different surfaces are quasi-isometrically distinct. We also show that the pants graphs of all but finitely many surfaces are quasi-isometrically rigid.