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Brumfield, G. and Hilden, Hugh Michael and Lozano Imízcoz, María Teresa and Montesinos Amilibia, José María and Ramírez Losada, E. and Short, H. and Tejada Cazorla, Juan Antonio and Toro, M.
(2008)
*Three manifolds as geometric branched coverings of the three sphere.*
Boletín de la Sociedad Matemática Mexicana. Tercera Serie, 14
(2).
pp. 263-282.
ISSN 1405-213X

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Official URL: http://www.smm.org.mx/boletinSMM/v14/14-2-5.pdf

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http://sociedadmatematicamexicana.org.mx/ | Organisation |

## Abstract

A finite covolume, discrete group of hyperbolic isometries U, acting on H3, is said to be universal if for every closed orientable 3-manifold M3 there is a finite index subgroup G of U so that M3=H3/G. It has been shown [H. M. Hilden et al., Invent. Math. 87 (1987), no. 3, 441–456;] that the orbifold group U of the Borromean rings with singular angle 90 degrees is universal and that H3/U=S3. In the present paper the authors construct a sequence of hyperbolic orbifold structures on S3 with orbifold groups Gi, i=1,…,4, such that G⊂G1⊂G2⊂G3⊂G4⊂U and they use this to obtain the following geometric branched covering space theorem: Let M3 be a closed orientable 3-manifold. Then there are finite index subgroups G⊂G1 of U such that M3=H3/G, S3=H3/G1 and the inclusion G→G1 induces a 3-fold simple branched covering M3→S3.

The group U acts as a group of isometries of hyperbolic 3-space H3 so that there is a tessellation of H3 by regular dodecahedra any one of which is a fundamental domain for U. The authors construct a closely related Euclidean crystallographic group Uˆ corresponding to a tessellation of E3 by cubes that are fundamental domains for Uˆ, and exhibit a homomorphism φ:U→Uˆ which defines a branched covering H3→E3 that respects the two tessellations. They classify the finite index subgroups of Uˆ, and use their pullback under φ to obtain the main result of the paper: For any positive integer n there is an index n subgroup of U generated by rotations.

Item Type: | Article |
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Uncontrolled Keywords: | branched covering; universal link; universal group |

Subjects: | Sciences > Mathematics > Topology |

ID Code: | 22006 |

Deposited On: | 19 Jun 2013 15:46 |

Last Modified: | 12 Dec 2018 15:13 |

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