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 –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 exist which have immersed –injective surfaces of negative Euler characteristic, but no such surface is virtually embedded (finitely covered by an embedded surface in some finite cover of ).
"Immersed and virtually embedded $\pi_1$–injective surfaces in graph manifolds." Algebr. Geom. Topol. 1 (1) 411 - 426, 2001. https://doi.org/10.2140/agt.2001.1.411