On finite index subgroups of a universal group

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.