Algebraic and Geometric Topology 1 (2001), paper no. 20, pages 411-426.

Immersed and virtually embedded pi_1-injective surfaces in graph manifolds

Walter D. Neumann

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

AMS subject classification. Primary: 57M10. Secondary: 57N10, 57R40, 57R42.

DOI: 10.2140/agt.2001.1.411

E-print: arXiv:math.GT/9901085

Submitted: 27 March 2001. Accepted: 6 July 2001. Published: 9 July 2001.

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

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