The BNS-invariant for Artin groups of circuit rank 2
type: Item
instance of: scholarly article
title: The BNS-invariant for Artin groups of circuit rank 2
zbMATH Open document ID: 1425.20019
author: Kisnney Almeida
published in: Journal of Group Theory
publication date: 2018-03-14T00:00:00Z
review text: The Artin group \(G\) with underlying finite simplicial graph \(\mathcal{G}\), with edges labeled by integers greater than one, is given by a finite presentation, with generators corresponding to the vertices \(V(\mathcal{G})\) of \(\mathcal{G}\) and relations given by \([u, v]_n = [v, u]_n\) for each edge \(\{u, v \}\) of \(\mathcal{G}\) with label \(n\), where \[ [a, b]_n = \underbrace{a b a b \ldots}_{n \text{ factors}} \text{ for all }a, b \in V(\mathcal{G}). \] If \(\mathcal{G}\) is connected and \(\pi_1 (\mathcal{G})\) is free of rank \(n\), then \(G\) is called an Artin group of \textit{circuit rank} \(n\). Let \(\mathcal{G}\) be a graph. A subgraph \(\mathcal{H}\) of \(\mathcal{G}\) is called \textit{dominant} if for each vertex \(v \in V(\mathcal{G}) \setminus V(\mathcal{H})\) there is an edge \(e \in E(\mathcal{G})\) such that \(\sigma(e) = v\) and \(\tau(e) \in V(\mathcal{H})\). Let \(\chi\) be a character of an Artin group \(G\) with underlying graph \(\mathcal{G}\). An edge \(e\) of \(\mathcal{G}\) is called \textit{dead} if \(e\) has an even label greater than 2 and \(\chi(\sigma(e)) = -\chi(\tau(e))\). Let \(G\) be an Artin group with underlying graph \(\mathcal{G}\) and let \(\chi : G \to \mathbb{R}\) be a non-zero real character of \(G\). Define \(\mathcal{L}_{\mathcal{F}} = \mathcal{L}_{\mathcal{F}}(\chi)\) as the full subgraph of \(\mathcal{G}\) generated by the vertices \(v \in V(\mathcal{G})\) such as \(\chi(v) \not= 0\). Define the \textit{living subgraph} \(\mathcal{L} = \mathcal{L}(\chi) \subset \mathcal{L}_{\mathcal{F}}\) as the subgraph obtained from \(\mathcal{L}_{\mathcal{F}}\) after removing the dead edges. The BNS-invariant or \(\Sigma^1\)-invariant is the first of a series of geometric invariants of finitely generated groups defined in the eighties that are deeply related to finiteness properties of their subgroups. \textit{J. Meier} [Proc. Lond. Math. Soc. (3) 74, No. 1, 151--173 (1997; Zbl 0869.20026)] calculated \(\Sigma^1(G)\) when…
reviewed by: Valeriy G. Bardakov
Mathematics Subject Classification ID: 20F36
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20J05
zbMATH DE Number: 6850106
zbMATH Keywords: Artin group
zbMATH Keywords: BNS-invariant
zbMATH Keywords: circuit rank 2
MaRDI profile type: Publication
full work available at URL: jgth-2017-0029
OpenAlex ID: W2755633315
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DOI: 10.1515/JGTH-2017-0029
description: scientific article; zbMATH DE number 6850106