Tits alternatives and low dimensional topology.
type: Item
instance of: scholarly article
title: Tits alternatives and low dimensional topology.
zbMATH Open document ID: 1056.57002
published in: Journal of the Mathematical Society of Japan
publication date: 2003-06-30T00:00:00Z
review text: The author gives results on groups of finite cohomological dimension in certain situations of interest in low dimensional topology. The motivation is Tits' theorem ``every finitely generated linear group is either virtually solvable or contains a nonabelian free subgroup''. The target groups in this paper are groups of cohomological dimension two, \(PD_3\)-groups, the fundamental groups of closed four-manifolds with Euler characteristic zero and two-knot groups. The paper also contains several applications on knot-groups. One of the main results is that if \(M\) is a closed 4-manifold with Euler characteristic 0 such that \(\pi=\pi(M)\) is locally virtually indicable and has no nonabelian free subgroup, and \(Z[\pi]\) is coherent, then either \(\pi\) is solvable and \(c.d.\pi=2\) or \(\pi\) is virtually \(Z^{2}\) or \(\pi\) has two ends or \(M\) is homeomorphic to an infrasolvmanifold.
reviewed by: Katsuya Yokoi
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 57N13
Mathematics Subject Classification ID: 57Q45
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 57M99
zbMATH DE Number: 1943315
zbMATH Keywords: coherent
zbMATH Keywords: 4-manifold
zbMATH Keywords: \(L^2\)-Betti number
zbMATH Keywords: minimal Seifert hypersurface
zbMATH Keywords: Poincaré duality group
zbMATH Keywords: solvable
zbMATH Keywords: Tits alternative
zbMATH Keywords: 2-knot
author: Jonathan A. Hillman
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/1191419121
OpenAlex ID: W1992921769
DOI: 10.2969/JMSJ/1191419121
description: scientific article; zbMATH DE number 1943315