We study the large-scale geometry of Weil–Petersson space, that is, Teichmüller space equipped with theWeil–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 Weil–Petersson 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 space.

中文翻译：

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Weil-Petersson指标的大规模等级和刚性

我们研究了Weil-Petersson空间的大规模几何，即配备了Weil-Petersson度量的Teichmüller空间。我们表明，这承认了特定等级的自然粗糙中值结构。假定这等于拟等距嵌入的欧几里德空间的最大尺寸，我们恢复Eskin，Masur和Rafi的结果，该结果给出了该空间的粗略秩。我们继续证明，除了有限的情况外，Weil-Petersson空间是准等距截然不同的，并且是准等距刚性的。特别地，这些空间之间的任何准等距是与等距的有界距离。根据布罗克定理，Weil–Petersson空间与裤子图等距拟等距，因此我们的结果同样适用于该空间。