Geometrisation of the mapping tori of Dehn twists on an infinite genus surfase

We study the conditions under which an infinite graph manifold $M$ carries a metric of nonpositive bounded curvature having finite volume. In the case where $M$ is the mapping torus of a collection of Dehn twists on an infinite genus surface and the graph of $M$ is linear (i.e., homeomorphic to a line or a ray) a complete list of all such manifolds is obtained.