Geometry & Topology

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

Jeffrey Brock and Howard Masur

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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.

Article information

Geom. Topol., Volume 12, Number 4 (2008), 2453-2495.

Received: 12 July 2007
Revised: 14 September 2008
Accepted: 10 July 2008
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30F60: Teichmüller theory [See also 32G15]
Secondary: 20F67: Hyperbolic groups and nonpositively curved groups

Teichmüller space Weil–Petersson geometry pants graph relative hyperbolicity


Brock, Jeffrey; Masur, Howard. Coarse and synthetic Weil–Petersson geometry: quasi-flats, geodesics and relative hyperbolicity. Geom. Topol. 12 (2008), no. 4, 2453--2495. doi:10.2140/gt.2008.12.2453.

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