Foundations of Hyperbolic Manifolds pp 652-714 | Cite as

# Geometric Orbifolds

## Abstract

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.

## Keywords

Discrete Group Orbit Space Cusp Point Geometric Space Local Isometry## Preview

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