Three manifolds as geometric branched coverings of the three sphere.



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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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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
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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