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The Orbifold Trick

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

Abstract

Suppose that O is a locally reflective 3-dimensional orbifold. We will assume that the interior of O admits a complete hyperbolic structure and \(|O|\) is orientable. The universal cover \(\mathbb{H}^3\) of int(O) has a partition into convex domains that are the components of the preimages of \(X_o {\sum}_{o} - \partial O\) under the universal cover \(\mathbb{H}^3 \rightarrow\) int(O). If C is the closure of one of those domains and \(G\,:= G_C\) is its stabilizer in \(\pi_1(O)\), then G C acts freely on C and \(|{\rm int}(O)| = C/G_C\) (see Section 6.5). Let \(R\,:= R_C\) be the subgroup in \(\pi_1(O)\) generated by reflections in the faces of C. Clearly, \(gRg^{-1} = R\) for every \(g \epsilon G\). According to Poincaré’s theorem on fundamental polyhedra, C is a fundamental domain for the action of R on \(\mathbb{H}^3\). We conclude that the fundamental group \(\pi_1(O)\) is generated by R and G and hence \(\pi_1(O) = R \ltimes G\). In particular, \(g \epsilon \pi_1(O)\) is orientation-reversing iff it is the product of an odd number of reflections in R and of an element of G.

Keywords

Boundary Component Finite Type Interior Vertex Isotopy Class Hyperbolic Structure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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