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arXiv · 0707.1469

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

Jeffrey Brock·Howard Masur
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Abstract

We analyze the coarse geometry of the Weil-Petersson metric on Teichmüller 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."

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Jeffrey Brock, Howard Masur. 2007-09-25. Coarse and synthetic Weil-Petersson geometry: quasi-flats, geodesics, and relative hyperbolicity. https://doi.org/10.2140/gt.2008.12.2453

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