W.Thurston raised the following question in 1976: Suppose that a compact 3-manifold M is not covered by (surface) or a torus bundle over . If and 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.
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Shicheng Wang, F. Yu, Covering degrees are determined by graph manifolds involved. Comment. Math. Helv. 74 (1999), no. 2, pp. 238–247DOI 10.1007/S000140050087