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 p : M → M ′ branched over L ′ . Moreover, if M ′ is a homology sphere then these topological types determine also the covering up to conjugacy.