Algebraic & Geometric Topology

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

Walter D Neumann

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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 π1–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 M3 exist which have immersed π1–injective surfaces of negative Euler characteristic, but no such surface is virtually embedded (finitely covered by an embedded surface in some finite cover of M3).

Article information

Algebr. Geom. Topol., Volume 1, Number 1 (2001), 411-426.

Received: 27 March 2001
Accepted: 6 July 2001
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M10: Covering spaces
Secondary: 57N10: Topology of general 3-manifolds [See also 57Mxx] 57R40: Embeddings 57R42: Immersions

$\pi_1$-injective surface graph manifold separable surface subgroup


Neumann, Walter D. Immersed and virtually embedded $\pi_1$–injective surfaces in graph manifolds. Algebr. Geom. Topol. 1 (2001), no. 1, 411--426. doi:10.2140/agt.2001.1.411.

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