Compactifying coverings of 3-manifolds
type: Item
instance of: scholarly article
title: Compactifying coverings of 3-manifolds
zbMATH Open document ID: 0873.57015
DOI: 10.1007/BF02566425
author: M. Mihalik
published in: Commentarii Mathematici Helvetici
publication date: 1997-11-02T00:00:00Z
full work available at URL: https://eudml.org/doc/140403
review text: Let \(M\) denote a compact, \(P^2\)-irreducible 3-manifold. \textit{J. Simon} [Mich. Math. J. 23(1976), 245-256 (1977; Zbl 0331.57002)] conjectured that any cover \(M_H\) of \(M\) corresponding to a finitely generated subgroup \(H\) of \(\pi_1(M)\) is a missing boundary manifold, meaning that \(M_H\) is homeomorphic to \(\overline M\smallsetminus K\), where \(\overline M\) is compact and \(K\subset\partial\overline M\). A key involves a geometric group theoretic concept, newly introduced here, of 1-tameness for pairs of groups. A revealing characterization establishes that, when \(H\) denotes a finitely generated subgroup of \(\pi_1(M)\), the pair \((\pi_1(M),H)\) is 1-tame if and only if, for each finite subcomplex \(C\) of \(M_H\) (the cover corresponding to \(H\)), \(\pi_1(M_H\smallsetminus C)\) is finitely generated. More importantly, in the same context \((\pi_1(M),H)\) is 1-tame if and only if \(M_H\) is a missing boundary manifold. The main result shows that if \(\pi_1(M)\) has an asynchronously bounded, almost prefix closed combing, and if \(H\) is a finitely generated subgroup of \(\pi_1(M)\) which is quasi-convex with respect to the combing, then \((\pi_1(M),H)\) is 1-tame, so \(M_H\) is a missing boundary manifold. It is known that any finitely presented group that is either word hyperbolic, automatic, or asynchronously automatic has an asynchronously bounded, almost prefix closed combing, as does the fundamental group of any compact 3-manifold satisfying the hypotheses of Thurston's geometrization conjecture.
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M10
Mathematics Subject Classification ID: 20F65
zbMATH DE Number: 949849
zbMATH Keywords: 3-manifold
zbMATH Keywords: missing boundary manifold
zbMATH Keywords: 1-tameness
zbMATH Keywords: asynchronously bounded
zbMATH Keywords: almost prefix closed combing
zbMATH Keywords: finitely presented group
zbMATH Keywords: word hyperbolic
zbMATH Keywords: automatic
reviewed by: Robert J. Daverman
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 949849