Train Tracks and the Gromov Boundary of the Complex of Curves

Providing each simplex in C(S) with the standard euclidean metric of side-length 1 equips the complex of curves with the structure of a geodesic metric space whose isometry group is just M̃g,m (except for the twice punctured torus). However, this metric space is not locally compact. Masur and Minsky [MM1] showed that nevertheless the geometry of C(S) can be understood quite explicitly. Namely, C(S) is hyperbolic of infinite diameter. Recall that for some δ > 0 a geodesic metric space is δ-hyperbolic in the sense of Gromov if it satisfies the δ-thin triangle condition: For every geodesic triangle with sides a, b, c the side c is contained in the δ-neighborhood of a ∪ b. Later Bowditch [B] gave a simplified proof of the result of Masur and Minsky which can also be used to compute explicit bounds for the hyperbolicity constant δ.