Virtual Haken 3-manifolds and Dehn filling
type: Item
instance of: scholarly article
title: Virtual Haken 3-manifolds and Dehn filling
zbMATH Open document ID: 0933.57014
DOI: 10.1016/S0040-9383(98)00061-5
author: Steven Boyer
author: Xingru Zhang
published in: Topology
publication date: 2000-03-12T00:00:00Z
review text: Let \(M\) be a compact, connected, orientable irreducible \(3\)-manifold whose boundary is a torus. A slope on \(\partial M\) is the isotopy class of an oriented, simple, essential loop in \(\partial M\) and gives rise to a closed \(3\)-manifold \(M(r)\) obtained by Dehn filling \(M\) with a solid torus so that a meridian of the torus has slope \(r\). Suppose \(M\) is hyperbolic in the sense that the interior of \(M\) admits a complete hyperbolic metric of finite volume. The distance between two slopes is their minimal geometric intersection number on \(\partial M\). The authors address the question, when is \(M(r)\) virtually Haken, i.e., finitely covered by a Haken manifold? The main results give conditions on one Dehn filling \(M(r_0)\) such that any other Dehn filling \(M(r)\) with the distance between \(r\) and \(r_0\) sufficiently large is virtually Haken. The virtual \(\mathbb Z\)-representability and virtual \(\mathbb Z\ast\mathbb Z\)-representability of \(\pi_1(M(r))\) is also discussed. Examples are given of \(M\) for which all but finitely many Dehn fillings are virtually Haken but not Haken.
reviewed by: Bruce Hughes
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57M27
zbMATH DE Number: 1356313
zbMATH Keywords: Haken 3-manifold
zbMATH Keywords: virtually Haken
MaRDI profile type: Publication
full work available at URL: s0040-9383(98)00061-5
OpenAlex ID: W2119978149
description: scientific article; zbMATH DE number 1356313