Coarse and synthetic Weil-Petersson geometry: quasi-flats, geodesics, and relative hyperbolicity

TL;DR

This paper investigates the coarse and synthetic geometry of the Weil-Petersson metric on Teichmüller space, revealing its relative hyperbolicity properties, the behavior of quasi-flats, and the structure of geodesics across different dimensions.

Contribution

It establishes the strong relative hyperbolicity of the pants graph in dimension 3 and characterizes the geometry and boundary structure of Weil-Petersson space in various dimensions.

Findings

In dimension 3, the pants graph is strongly relatively hyperbolic.

No non-trivial collection of subsets makes the space strongly relatively hyperbolic in higher dimensions.

Complete description of the CAT(0)-boundary via curve-hierarchies and boundary laminations.

Abstract

We analyze the coarse geometry of the Weil-Petersson metric on Teichm\"uller space, focusing on applications to its synthetic geometry (in particular the behavior of geodesics). We settle the question of the strong relative hyperbolicity of the Weil-Petersson metric via consideration of its coarse quasi-isometric model, the "pants graph." We show that in dimension~3 the pants graph is strongly relatively hyperbolic with respect to naturally defined product regions and show any quasi-flat lies a bounded distance from a single product. For all higher dimensions there is no non-trivial collection of subsets with respect to which it strongly relatively hyperbolic; this extends a theorem of [BDM] in dimension 6 and higher into the intermediate range (it is hyperbolic if and only if the dimension is 1 or 2 [BF]). Stability and relative stability of quasi-geodesics in dimensions up through 3 provide for a strong understanding of the behavior of geodesics and a complete description of the CAT(0)-boundary of the Weil-Petersson metric via curve-hierarchies and their associated "boundary laminations."