Augmented group systems and shifts of finite type
type: Item
instance of: scholarly article
title: Augmented group systems and shifts of finite type
zbMATH Open document ID: 0899.20011
DOI: 10.1007/BF02761041
author: Daniel S. Silver
author: Susan G. Williams
published in: Israel Journal of Mathematics
publication date: 1998-09-24T00:00:00Z
review text: Given an epimorphism \(\chi\colon G\to\mathbb{Z}\) of the finitely generated group \(G\) onto the integers, the authors investigate \(K_\chi=\text{kernel} \chi\). The situation arises for \(G\) a knot group and \(K_\chi\) its commutator subgroup; it is well known that \(K_\chi\) might or might not be finitely generated. The main result is: \(K_\chi\) contains finitely many subgroups \(H\) with index \([K_\chi:H]\leq r\) for any \(r<\infty\), if \(\chi\) represents an element of \((\Hom(G,\mathbb{R})-\{0\})/\mathbb{R}_+\) which belongs to the Bieri-Neumann-Strebel subset \(\Sigma\). The argument uses the sets \(\phi_r\) of representations \(\rho\colon\text{Kernel} \chi\to S_r\) (= symmetric group) which can be related to the subgroups \(H\) by covering theory, and shifts \[ \sigma_x\colon\phi_r\to\phi_r,\quad(\sigma_x\rho)a=\rho(x^{-1}ax)\text{ for }x\in\chi^{-1}(1). \] From the shifts an entropy invariant can be derived with application to knot theory. A corollary to the main theorem states that \(K_\chi\) is Hopfian if \(G\) is residually finite and \(\chi\) in \(\Sigma\). As a counterexample the authors show that the commutator subgroup of the group of the knot \(5_2\) has uncountably many subgroups of index \(r>3\).
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57Q45
zbMATH DE Number: 952565
zbMATH Keywords: augmented group systems
zbMATH Keywords: finitely generated groups
zbMATH Keywords: knot groups
zbMATH Keywords: commutator subgroups
zbMATH Keywords: Bieri-Neumann-Strebel subsets
zbMATH Keywords: covering theory
zbMATH Keywords: shifts
reviewed by: Gerhard Burde
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02761041
OpenAlex ID: W2018969467
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description: scientific article; zbMATH DE number 952565