Covering degrees are determined by graph manifolds involved

W.Thurston raised the following question in 1976: Suppose that a compact 3-manifold M is not covered by (surface) x S-1 or a torus bundle over S-1. If M-1 and M-2 are two homeomorphic finite covering spaces of M, do they have the same covering degree? For so called geometric 3-manifolds (a famous conjecture is that all compact orientable 3-manifolds are geometric), it is known that the answer is affirmative if M is not a non-trivial graph manifold. In this paper, we prove that the answer for non-trivial graph manifolds is also affirmative. Hence the answer for the Thurston's question is complete for geometric 3-manifolds. Some properties of S-manifold groups are also derived.MathematicsSCI(E)7ARTICLE2238-2477