A structure theorem for irreducible open graph 3-manifolds

Graph manifolds are a class of compact, orientable 3-manifolds introduced in 1967 by Waldhausen as a generalization of Seifert fibered 3-manifolds. From the point of view of Thurston's geometrization program, graph manifolds are exactly the compact, orientable 3-manifolds without any hyperbolic piece in their geometric decomposition.

In this article we consider a generalization of the notion of graph manifold that includes some noncompact 3-manifolds. We prove a structure theorem for irreducible open graph manifolds in the form of a canonical 'reduced' decomposition along embedded, incompressible 2-tori.