## Algebraic & Geometric Topology

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

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

#### 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 ${M}^{3}$).

#### 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

**Subjects**

Primary: 57M10: Covering spaces

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

**Keywords**

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

#### 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