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Teichmüller space is not Gromov hyperbolic. (English) Zbl 0878.32015
Any finite dimensional Teichmüller space, equipped with the Teichmüller metric, is a straight (Finsler) space in the sense of Busemann. However, it is not negatively curved in the Busemann sense. Gromov introduced a notion of hyperbolicity that captures some of the features of negatively curved spaces; it is shown in this article that the Teichmüller spaces fail to be hyperbolic in Gromov’s sense also. The method is to study a certain family of geodesic triangles whose vertices are represented by biholomorphically equivalent Riemann surfaces, and utilizing Jenkins-Strebel differentials.
Reviewer: S.Nag (Madras)

32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
30F60 Teichmüller theory for Riemann surfaces
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
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