Immersed essential surfaces in hyperbolic 3-manifolds
type: Item
instance of: scholarly article
title: Immersed essential surfaces in hyperbolic 3-manifolds
zbMATH Open document ID: 1019.57009
author: Tao Li
published in: Communications in Analysis and Geometry
publication date: 2003-04-07T00:00:00Z
review text: Many of the central questions regarding 3-manifolds are known for the class of Haken manifolds (e.g. results of Waldhausen, Haken and Thurston). A 3-manifold is Haken if it is irreducible and contains an embedded \(\pi_1\)-injective surface. Since most questions of interest have been reduced to irreducible manifolds, this assumption is not in general problematic; however, there are known constructions, which are quite general, of manifolds not containing a \(\pi_1\)-injective surface. It is therefore of great interest to find structures on irreducible-3-manifolds that generalize embedded \(\pi_1\)-injective surfaces. The current paper is concerned with one such possibility: \(\pi_1\)-injective surfaces (but not necessarily embedded). \textit{B. Freedman} and \textit{M. H. Freedman} [Topology 37, 133-147 (1998; Zbl 0896.57012)] constructed immersed \(\pi_1\)-injective surfaces in a manifold with boundary torus (say \(X)\) by first considering a \(\pi_1\)-injective surface with boundary (say \(F)\). They proved that either \(F\) is a fiber in a fibration over \(S^1\) or, after tubing \(F\) to itself along an annulus that runs along the boundary many times, one obtains a closed surface that compresses down to a \(\pi_1\)-injective surface. According to \textit{M. Culler} and \textit{P. B. Shalen} [Invest. Math. 75, 537-545 (1984; Zbl 0542.57011)] (see also [\textit{M. Culler}, \textit{Mc A. Gordon}, \textit{J. Luecke} and \textit{P. B. Shalen}, Ann. Math. (2) 127, 663 (1988; Zbl 0645.57008)]) any manifold \(X\) contains bounded \(\pi_1\)-injective surfaces that are not fibers over \(S^1\) for at least 2 different boundary slopes. Their construction was used and improved in many papers, including the paper under review. Given a manifold with boundary torus one can obtain a closed manifold by attaching a solid torus to the boundary; this is called Dehn filling. It is well known that whenever \(X\) is hyperbolic infinitely many distinct manifolds are obtained. If, in addi…
reviewed by: Yo'av Rieck
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M50
zbMATH DE Number: 1893846
zbMATH Keywords: immersed surface
zbMATH Keywords: Dehn surgery
zbMATH Keywords: 3-manifolds
zbMATH Keywords: Haken manifolds
MaRDI profile type: Publication
full work available at URL: cag.2002.v10.n2.a2
OpenAlex ID: W39892798
DOI: 10.4310/CAG.2002.V10.N2.A2
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description: scientific article; zbMATH DE number 1893846