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The Weil-Petersson geometry of the five-times punctured sphere. (English) Zbl 1103.57012
Minsky, Yair (ed.) et al., Spaces of Kleinian groups. Proceedings of the programme ‘Spaces of Kleinian groups and hyperbolic 3-manifolds’, Cambridge, UK, July 21–August 15, 2003. Cambridge: Cambridge University Press (ISBN 0-521-61797-9/pbk). London Mathematical Society Lecture Note Series 329, 219-231 (2006).
In the paper [Am. J. Math. 128, 1–22 (2006; Zbl 1092.32008)], J. Brock and B. Farb proved that the Teichmüller space of a compact surface of genus $$g$$ and $$n$$ boundary components is Gromov-hyperbolic if and only if $$3g - 3 + n\leq 2$$. When $$3g - 3 + n \geq 3$$, the Weil-Petersson metric has higher rank in the sense of Gromov.
In the paper under review, the author gives another proof of the fact that the Teichmüller spaces of the five-punctured sphere and of the twice-punctured torus (where $$3g - 3 + n =2$$), equipped with Weil-Petersson metric, are Gromov-hyperbolic. The author works in the Weil-Petersson completion of Teichmüller space, which is a CAT(0) space. The proof of his result, unlike the proof by Brock and Farb, does not make use of the geometry of the curve complex, but relies on geometric properties of CAT(0) spaces. As a matter of fact, the author’s result, given the fact that the Teichmüller space completion is quasi-isometric to the curve complex, provides another proof that the curve complex is Gromov hyperbolic in the case where the surface is a five-punctured sphere or a twice-punctured torus.
For the entire collection see [Zbl 1089.30004].

##### MSC:
 57M50 General geometric structures on low-dimensional manifolds 51K10 Synthetic differential geometry 30F60 Teichmüller theory for Riemann surfaces 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
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