Immersed and virtually embedded π1-injective surfaces in graph manifolds

We show that many 3-manifold groups have no nonabelian surface subgroups. For example, any link of an isolated complex surface singularity has this property. In fact, we determine the exact class of closed graph-manifolds which have no immersed 1 -injective surface of negative Euler characteristic. We also determine the class of closed graph manifolds which have no nite cover containing an embedded such surface. This is a larger class. Thus, manifolds M exist which have immersed 1 -injective surfaces of negative Euler characteristic, but no such surface is virtually embedded ( nitely covered by an embedded surface in some nite cover of M ). AMS Classi cation 57M10; 57N10, 57R40, 57R42