Anosov flows on graph manifolds in the sense of Waldhausen

This paper is devoted to the study of a wide class of Anosov flows on graph manifolds. We establish a general result about embeddings of Seifert manifolds in 3-dimensional manifolds admitting a product Anosov flow, generalizing a previous result of E. Ghys. We show that up to isotopy the restriction of the one-dimensional foliation defined by the flow to the image of this embedding is topologically conjugate to a piece of a geodesic flow outside a finite number of periodic orbits. We show more: the conjugacy can be chosen such that it respects the restrictions of the weak foliations. We next give a topological characterization of the examples of Handel-Thurston. Essentially, they are the unique Anosov flows on graph manifolds such that no periodic orbit is freely homotopic to the fiber of some embedded Seifert manifold in the graph manifold. Eventually, we exhibit the first known examples of graph manifolds which are not Seifert spaces nor torus bundles over the circle, whose fundamental groups are of exponential growth and admitting no Anosov flow.