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

Walter D Neumann

Algebraic & Geometric Topology 1 (2001) 411–426

arXiv: math.GT/9901085

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).

Keywords
$\pi_1$-injective surface, graph manifold, separable, surface subgroup
Mathematical Subject Classification 2000
Primary: 57M10
Secondary: 57N10, 57R40, 57R42
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Publication
Received: 27 March 2001
Accepted: 6 July 2001
Published: 9 July 2001
Authors
Walter D Neumann
Department of Mathematics
Barnard College
Columbia University
New York NY 10027
USA