Three manifolds as geometric branched coverings of the three sphere.

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.