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

Abstract

We analyze the coarse geometry of the Weil–Petersson metric on Teichmuller 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 nontrivial collection of subsets with respect to which it strongly relatively hyperbolic; this extends a theorem of Behrstock, Drutu and Mosher [submitted] in dimension 6 and higher into the intermediate range (it is hyperbolic if and only if the dimension is 1 or 2 by Brock and Farb [Amer. J. Math. 128 (2006) 1-22]). 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.