Let \(L= K_1\cup K_2\) be an oriented link in the 3-sphere \(S^3\) with two components. We denote by \(m_1\) resp. \(m_2\) the oriented meridian of \(K_1\) resp. \(K_2\). For a fixed integer \(n\geq 2\) and any integer \(k\) such that \(\text{gcd} (k,n)=1\) we consider the homomorphism \(\psi_{n,k}: \pi_1 (S^3- L)\to \mathbb{Z}/ n\mathbb{Z}\) defined by \(\psi_{n,k} (m_1)= \overline {1}\), \(\psi_{n,k} (m_2)= \overline {k}\). Denote by \(M_{n,k}\) the cyclic branched covering corresponding to the kernel of \(\psi_{n,k}\). Take, for example, the Hopf link then \(M_{n,k}\) is the lens space \(L(n, k)\). From the classification of lens spaces it follows that \(M_{n,k}\) and \(M_{n,k'}\) are homeomorphic if and only if \(k\equiv k' \pmod n\) or \(kk'\equiv \pm 1\pmod n\).
The aim of the paper under review is to prove an analogous result for hyperbolic links. More precisely, let \(L\subset S^3\) be a hyperbolic link and let \(n\geq 2\) be an integer. The main theorem relates the classification up to isometry or homeomorphism of the manifolds \(M_{n,k}\) to the symmetry group of the link \(L\). In various cases it allows a complete classification of the manifolds \(M_{n,k}\). As an example the cyclic branched coverings of the Whitehead link are classified.