On cyclic branched coverings of hyperbolic links

Abstract

Two of the main methods for the construction of closed orientable 3-manifolds, and in particular of hyperbolic 3-manifolds, are surgery on links and branched coverings of links. If the link is hyperbolic, i.e., has hyperbolic complement, by results of Thurston most of the resulting 3-manifolds are hyperbolic. In the present paper, for a fixed integer n ⩾ 2, we consider hyperbolic 3-manifolds M n, k which are cyclic n-fold branched coverings of a hyperbolic link with two components. Our main theorem relates the classification up to isometry or homeomorphism of these manifolds to the symmetry group of the link and allows a complete classification of these manifolds in various cases; as an example, we consider cyclic branched coverings of the Whitehead link. The classification resembles the classification of lens spaces which are the cyclic branched coverings of the Hopf link (which is not hyperbolic, however); it generalizes to links with more than two components. For the proof, which consists in a mixture of geometric and algebraic arguments, we extend methods used in [5] in a special case.