Hyperbolic Manifolds and Discrete Groups pp 403-416 | Cite as

# The Orbifold Trick

## 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## Preview

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