Canonical splittings of groups and 3-manifolds
type: Item
instance of: scholarly article
title: Canonical splittings of groups and 3-manifolds
zbMATH Open document ID: 0981.57002
DOI: 10.1090/S0002-9947-01-02871-9
author: Peter Scott
author: Gadde A. Swarup
published in: Transactions of the American Mathematical Society
publication date: 2001-10-21T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0107232
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 1663174
zbMATH Keywords: 3-manifold
zbMATH Keywords: characteristic submanifold
zbMATH Keywords: Deformation Theorem
zbMATH Keywords: ends of groups
zbMATH Keywords: JSJ decomposition
MaRDI profile type: Publication
arXiv ID: math/0107232
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OpenAlex ID: W1616942760
review text: The standard JSJ-decomposition of an orientable Haken 3-manifold is described in terms that can be translated into group theory and a purely algebraic description of it is given. This leads to a natural algebraic proof of Johannson's Deformation Theorem [\textit{K. Johannson}, Homotopy equivalences of 3-manifolds with boundaries, Lect. Notes Math. 761 (1979; Zbl 0412.57007)]. In the first section, the authors give the background material on 3-manifolds and group theory. In the second section analogues of canonic annuli and tori are studied, namely canonical \(\mathbb{Z}\) or \(\mathbb{Z}\times \mathbb{Z}\) splittings of \(G=\pi _{1}(M)\). Since this characterization is algebraic, it follows that the JSJ-systems of any two Haken manifolds with isomorphic fundamental groups correspond. The last section deals with the result that we mentioned above.
reviewed by: Viorel Mihai Gontineac
description: scientific article; zbMATH DE number 1663174