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On finite index subgroups of a universal group

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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) On finite index subgroups of a universal group. Boletín de la Sociedad Matemática Mexicana. Tercera Serie, 14 (2). pp. 283-302. ISSN 1405-213X

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Official URL: http://www.smm.org.mx/boletinSMM/v14/14-2-6.pdf




Abstract

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, i.e. for every closed orientable 3-manifold M3 there is a finite index subgroup G of U such that M3=H3/G. Since the fundamental group of M3 is the quotient of G modulo the subgroup generated by rotations, one would like to classify the finite index subgroups of U. In this paper, the authors begin the classification of the finite index subgroups that are generated by rotations.
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:3-manifold; orbifold; branched covering; universal link; universal group
Subjects:Sciences > Mathematics > Topology
ID Code:21872
Deposited On:14 Jun 2013 17:58
Last Modified:12 Dec 2018 15:13

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