  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 pi_1-injective surface of negative Euler characteristic. We also
determine the class of closed graph manifolds which have no finite cover
containing an embedded such surface. This is a larger class. Thus, manifolds
M^3 exist which have immersed pi_1-injective surfaces of negative Euler
characteristic, but no such surface is virtually embedded (finitely covered by
an embedded surface in some finite cover of M^3).
