When closed graph manifolds are finitely covered by surface bundles over S 1

The problem of deciding whether a graph manifold is finitely covered by a surface bundle over the circle is discussed in this paper. A necessary and sufficient condition in term of the solutions of a certain matrix equation is obtained, as well as a necessary condition which is easy to compute. Our results sharpen and extend the earlier results of J. Leucke-Y. Wu, W. Neumann, and S. Wang-F. Yu in this topic.