#### Volume 1, issue 1 (2001)

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Immersed and virtually embedded $\pi_1$–injective surfaces in graph manifolds

### Walter D Neumann

Algebraic & Geometric Topology 1 (2001) 411–426
 arXiv: math.GT/9901085
##### Abstract

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 ${M}^{3}$).

##### Keywords
$\pi_1$-injective surface, graph manifold, separable, surface subgroup
##### Mathematical Subject Classification 2000
Primary: 57M10
Secondary: 57N10, 57R40, 57R42