Geometric Orbifolds

  • John G. Ratcliffe
Part of the Graduate Texts in Mathematics book series (GTM, volume 149)


In this chapter, we study the geometry of geometric orbifolds. We begin by studying the geometry of an orbit space of a discrete group of isometries of a geometric space. In Section 13.2, we study orbifolds modeled on a geometric space X via a group G of similarities of X. Such an orbifold is called an (X, G)-orbifold. In particular, if Γ is a discrete group of isometries of X, then the orbit space X/Γ is an (X, G)-orbifold for any group G of similarities of X containing Γ. In Section 13.3, we study the role of metric completeness in the theory of (X, G)-orbifolds. In particular, we prove that if M is a complete (X, G)-orbifold, with X simply connected, then there is a discrete subgroup Γ of G of isometries of X such that M is isometric to X/Γ. In Section 13.4, we prove the gluing theorem for geometric orbifolds. The chapter ends with a proof of Poincaré’s fundamental polyhedron theorem.


Discrete Group Orbit Space Cusp Point Geometric Space Local Isometry 
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Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • John G. Ratcliffe
    • 1
  1. 1.Department of MathematicsVanderbilt UniversityNashvilleUSA

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