Book contents
- Frontmatter
- Contents
- Preface
- Preface 2005
- PART I NOTES ON NOTES OF THURSTON
- A New Foreword
- Introduction to Part I
- Chapter 1.1 (G, X)-structures
- Chapter I.2 Hyperbolic structures
- Chapter I.3 Spaces of hyberbolic manifolds
- Chapter I.4 Laminations
- Chapter I.5 Pleated surfaces
- PART II CONVEX HULLS IN HYPERBOLIC SPACE, A THEOREM OF SULLIVAN, AND MEASURED PLEATED SURFACES
- PART III EARTHQUAKES IN 2-DIMENSIONAL HYPERBOLIC GEOMETRY
- PART IV LECTURES ON MEASURES ON LIMIT SETS OF KLEINIAN GROUPS
Chapter I.5 - Pleated surfaces
Published online by Cambridge University Press: 05 November 2011
- Frontmatter
- Contents
- Preface
- Preface 2005
- PART I NOTES ON NOTES OF THURSTON
- A New Foreword
- Introduction to Part I
- Chapter 1.1 (G, X)-structures
- Chapter I.2 Hyperbolic structures
- Chapter I.3 Spaces of hyberbolic manifolds
- Chapter I.4 Laminations
- Chapter I.5 Pleated surfaces
- PART II CONVEX HULLS IN HYPERBOLIC SPACE, A THEOREM OF SULLIVAN, AND MEASURED PLEATED SURFACES
- PART III EARTHQUAKES IN 2-DIMENSIONAL HYPERBOLIC GEOMETRY
- PART IV LECTURES ON MEASURES ON LIMIT SETS OF KLEINIAN GROUPS
Summary
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).
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- Fundamentals of Hyperbolic ManifoldsSelected Expositions, pp. 89 - 116Publisher: Cambridge University PressPrint publication year: 2006