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A geometric approach to the complex of curves on a surface. (English) Zbl 0937.30027
Kojima, Sadayoshi (ed.) et al., Topology and Teichmüller spaces. Proceedings of the 37th Taniguchi symposium, Katinkulta, Finland, July, 24-28, 1995. Singapore: World Scientific. 149-158 (1996).
Let \(S\) be a finite genus surface with finitely many punctures and \(K(S)\) be the 1-skeleton of the simplicial complex of curves introduced by W. J. Harvey [Ann. Math. Stud. 97, 245-251 (1981; Zbl 0461.30036)]. In a forth-coming paper by H. Masur and the author the following theorem will be proved: The complex \(K(S)\) endowed with the metric giving length 1 to edges is \(\delta\)-hyperbolic for some \(\delta(S)\geq 0\). This article is an exposition of the ideas and motivations behind this theorem. A proof is sketched and some remaining questions and future plans are noted.
For the entire collection see [Zbl 0901.00037].

30F60 Teichmüller theory for Riemann surfaces
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)