#### 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.

Notes on file formats
Walter D. Neumann

Department of Mathematics, Barnard College, Columbia University

New York, NY 10027, USA

Email: neumann@math.columbia.edu

AGT home page

## Archival Version

**These pages are not updated anymore.
They reflect the state of
.
For the current production of this journal, please refer to
http://msp.warwick.ac.uk/.
**