Large-scale rank and rigidity of the Weil–Petersson metric

We study the large-scale geometry of Weil-Petersson space, that is, Teichmüller space equipped with the Weil-Petersson metric. We show that this admits a natural coarse median structure of a specific rank. Given that this is equal to the maximal dimension of a quasi-isometrically embedded euclidean space, we recover a result of Eskin, Masur and Rafi which gives the coarse rank of the space. We go on to show that, apart from finitely many cases, the WeilPetersson spaces are quasi-isometrically distinct, and quasi-isometrically rigid. In particular, any quasi-isometry between such spaces is a bounded distance from an isometry. By a theorem of Brock, Weil-Petersson space is equivariantly quasi-isometric to the pants graph, so our results apply equally well to that