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].