Finiteness of Mapping Degrees and Psl(2,r)-volume on Graph Manifolds

For given closed orientable 3-manifolds M and N let D(M,N) be the set of mapping degrees from M to N . We address the problem: For which N , D(M,N) is finite for all M? The answer is known in Thurston’s picture of closed orientable irreducible 3-manifolds unless the target is a non-trivial graph manifold. We prove that for each closed non-trivial graph manifold N , D(M,N) is finite for all graph manifold M . The proof uses a recently developed standard forms of maps between graph manifolds and the estimation of the g PSL(2,R)-volume for certain class of graph manifolds.