The Orbifold Trick

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


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.


Boundary Component Finite Type Interior Vertex Isotopy Class Hyperbolic Structure 
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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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