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.
