Covering degrees are determined by graph manifolds involved

Abstract

W.Thurston raised the following question in 1976: Suppose that a compact 3-manifold M is not covered by (surface) $\times S^1$ 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 3-manifold groups are also derived.