Spheres in the Curve Complex

The curve graph (or curve complex) C(S) associated to a surface S of finite type is a locally infinite combinatorial object that encodes topological information about the surface through intersection patterns of simple closed curves. It is known to be δ-hyperbolic [5], a property that is often described by saying that a space is “coarsely a tree.” To be precise, there exists δ such that for any geodesic triangle, each side is in the δ-neighborhood of the union of the other two sides. In this note, we will investigate the finer metric properties of the curve graph by considering the geometry of spheres; specifically, we will study the average distance between pairs of points on Sr(α), the sphere of radius r centered at α. To make sense of the idea of averaging, we will develop a definition of null and generic sets in §3 that is compatible with the topological structure of the curve graph. Given a family of probability measures μr on the spheres Sr(x) in a metric space (X, d), let E(X) = E(X,x, d, {μr}) be the normalized average distance between points on large spheres: