Orders and actions of branched coverings of hyperbolic links (Q1019147)

Let \(M\), \(M'\) be compact oriented 3-manifolds. Let \(L'\subset M'\) be a nonempty link and \(G\) a group that acts orientation-preservingly on \(M\) such that the natural projection \(p:M\to M/G\cong M'\) is a branched covering with branch set \(L'\). The pre-image of \(L'\) in \(M\) is a link \(L\). If the stabilizer of each point \(x\in L = p^{-1}(L')\) equals the covering group \(G\) then \(p\) is called \textit{strongly branched}. The author proves the following: Let \(M\), \(M'\) be compact oriented 3-manifolds whose boundary is a (possible empty) disjoint union of tori. Let \(L'\subset M'\) be a prime link. If the exterior of \(L'\) in \(M'\) is irreducible and its JSJ (Jaco-Shalen-Johannson) decomposition contains a hyperbolic piece, then there exists at most one positive integer \(d\) for which \(M\) is a \(d\)-fold covering of \(M'\), strongly branched over \(L'\). Moreover if \(N'\) is an integral homology 3-sphere and \(L'\) is a hyperbolic link, then for any manifold \(M\) any two cyclic coverings \(p_1,p_2:M\to M'\) over \(L'\), of prime degrees, are conjugate.