Coarse and synthetic Weil-Petersson geometry: Quasi-flats, geodesics and relative hyperbolicity. (English) Zbl 1176.30096

The authors study the coarse geometry of the Weil-Petersson metric on Teichmüller space, focusing on applications to its sinthetic geometry (in particular the behavior of geodesics). They define and study the quasi-flats in metric spaces, the boundary of the Weil-Petersson metric and thickness and relative hyperbolicity. After defining the quasi-geodesics, they obtain a relative stability of quasi-geodesics. Then they obtain some applications, giving a finer structure of the \(CAT(0)\) boundary of the Weil-Petersson metric for \(S\) with \(\zeta(S)\leq 3\). Finally, they exhibit the thickness of the pants graph \(P(S)\) and thence the Weil-Petersson metric for \(S\) with \(\zeta(S)=4\) and \(5\) and discuss how it follows naturally that the Weil-Petersson metric cannot be strongly relatively hyperbolic with respect to any collection of co-infinite subsets for \(\zeta(S)\geq 4\).


30F60 Teichmüller theory for Riemann surfaces
20F67 Hyperbolic groups and nonpositively curved groups
53C22 Geodesics in global differential geometry
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