Introduction

We now discuss pleated surfaces, which are a basic tool in Thurston's analysis of hyperbolic structures on 3-manifolds. See Section 8.8 of Thurston (1979); there, pleated surfaces are called uncrumpled surfaces. Recall from definition under Section I.1.3.3 (Isometric map) that an isometric map takes rectifiable paths to rectifiable paths of the same length.

Definition. A map f: M → N from a manifold M to a second manifold N is said to be homotopically incompressible if the induced map f*: π1(S) → π1(M) is injective.

Definition. A pleated surface in a hyperbolic 3-manifold M is a complete hyperbolic surface S together with an isometric map f : S → M such that every point s ∈ S is in the interior of some geodesic arc which is mapped by f to a geodesic arc in M. We shall also require that f be homotopically incompressible.

Note that this definition implies that a pleated surface f maps cusps to cusps since horocyclic loops on S are arbitrarily short and f is isometric and homotopically incompressible.

Definition. If (S, f) is a pleated surface, then we define its pleating locus to be those points of S contained in the interior of one and only one geodesic arc which is mapped by f to a geodesic arc.

An example of a pleated surface is the boundary of the convex core (see Part II).