The curve complex and its relatives

As we have seen in the previous lecture, Teichmüller space T (S) for a surface of finite type S = Sg,n (genus g and n punctures or boundary components) is a parameter space for various types of structures that can be imposed on the same topological surface S. I’ll adopt the view that T (S) is the space of metrics on S, marked with a set of generators for the fundamental group, up to (measurable) conformal equivalence: in that way, each point represents a conformal class of metrics. Let us use ξ = ξ(S) = 3g − 3 + n as a measure of the topological complexity of S; then when ξ ≥ 1, there are hyperbolic metrics on S. Every conformal class contains many singular flat metrics and, by uniformization, exactly one Poincaré metric (constant curvature −1). Thus we can also identify points of T (S) with their hyperbolic representatives.