Geometry of the Complex of Curves Ii: Hierarchical Structure

In this paper we continue our geometric study of Harvey’s Complex of Curves [12], a finite dimensional and locally infinite complex C(S) associated to a surface S, which admits an action by the mapping class group Mod(S). The geometry and combinatorics of C(S) can be applied to study grouptheoretic properties of Mod(S), and the geometry of Kleinian representations of π1(S). In [19] we showed that, endowed with a natural metric, C(S) is an infinite diameter δ-hyperbolic space in all but a small number of trivial cases (see Section 2 for precise definitions). This result suggests that one try to apply the techniques of hyperbolic spaces and groups to study C(S) and its Mod(S)-action, considering for example such questions as the word problem, conjugacy problem and quasi-isometric rigidity. The barrier to doing this is that the complex is locally infinite, and hence the distance bounds one obtains in a typical geometric argument give little a-priori information. Our goal in this paper is to develop tools for lifting this barrier. The organizing philosophy is roughly this: Links of vertices in C(S) are themselves