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.