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 $π_{1}$ –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 $π_{1}$ –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

References

Forward citations

Publication

Received: 27 March 2001

Accepted: 6 July 2001

Published: 9 July 2001

Authors

Walter D Neumann
Department of Mathematics Barnard College Columbia University New York NY 10027 USA

Volume 1, issue 1 (2001)

Walter D Neumann