Pleated Surfaces and Ends of Hyperbolic Manifolds

  • Michael Kapovich
Part of the Modern Birkhäuser Classics book series (MBC)


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 VS be a finite subset. A “triangulationT of S based at VP 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 vP), (2) no arc is isotopic to a point in \(\hat{S}\) (mod VP); (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.


Fundamental Group Hyperbolic Manifold Closed Geodesic Hyperbolic Surface Pleated Surface 
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Copyright information

© Birkhäuser Boston 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, DavisDavisU.S.A.

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