## Algebraic & Geometric Topology

### 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 $π1$–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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513882600

Digital Object Identifier
doi:10.2140/agt.2001.1.411

Mathematical Reviews number (MathSciNet)
MR1852764

Zentralblatt MATH identifier
0979.57007

#### Citation

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. https://projecteuclid.org/euclid.agt/1513882600

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