Orders and actions of branched coverings of hyperbolic links

Let M, M′ be compact oriented 3-manifolds and L′ a link in M′ whose exterior has positive Gromov norm. We prove that the topological types of M and (M′,L′) determine the degree of a strongly cyclic covering branched over L′. Moreover, if M′ is a homology sphere then these topological types determine also the covering up to conjugacy.