Determining knots and links by cyclic branched coverings

It is well known that different knots or links in the 3-sphere can have homeomorphic n-fold cyclic branched coverings. We consider the following problem: for which values of n is a knot or link determined by its n-fold cyclic branched covering? We consider the class of hyperbolic resp. 2 pi/n-hyperbolic links. The isometry or symmetry groups of such links are finite, and their cyclic n-fold branched coverings are hyperbolic 3-manifolds. Our main result states that if n does not divide the order of the finite symmetry group of such a link, then the link is determined by its n-fold cyclic branched covering. In a sense, the result is best possible; the key argument of its proof is algebraic using some basic result about finite p-groups. The main result applies, for example, to the cyclic branched coverings of the 2-bridge links; in particular, it gives a classification of the maximally symmetric D-6 -manifolds which are exactly the 3-fold branched coverings of the 2-bridge links.