Hyperbolic Manifolds and Discrete Groups pp 351-368 | Cite as

# Pleated Surfaces and Ends of Hyperbolic Manifolds

## Abstract

Consider a smooth connected closed surface \(\hat{S}\) and a finite subset *P* = {*p1,…,pm*} in \(\hat{S}\) such that the punctured surface *S* := \(\hat{S} - P\) is of hyperbolic type. Pick a complete hyperbolic structure on *S*. Let *V* ∪ *S* be a finite subset. A “*triangulation*” *T* of *S* based at *V* ∪ *P* is a finite collection ∆ of smooth compact disjoint arcs in \(\hat{S}\) with endpoints in *V* Ȫ *P* such that (1) no two arcs are isotopic in \(\hat{S}\) (mod *v* ∪ *P*), (2) no arc is isotopic to a point in \(\hat{S}\) (mod *V* ∪ *P*); (3) this system is a maximal system of arcs with properties (1) and (2). Note that such *“triangulation”* does not have to be a triangulation in the usual sense. In our applications, *v* will be either a single vertex or an empty set. The points in *V* are called *vertices* and the points in *P* are called the *ideal vertices* of the *triangulation*. The *edges* of the triangulation are the arcs in ∆. We define the *triangles* of the triangulation to be metric completions of the completions of the components of *S*–∆. The maximality of *T* implies that each triangle τ has three edges and three vertices (Some of which may be ideal). The interior of each edge of τ maps bijectively to an edge of *T*. However, distinct edges may have equal images.

## Keywords

Fundamental Group Hyperbolic Manifold Closed Geodesic Hyperbolic Surface Pleated Surface## Preview

Unable to display preview. Download preview PDF.