Locally non-spherical Artin groups
type: Item
instance of: scholarly article
title: Locally non-spherical Artin groups
zbMATH Open document ID: 0901.20025
author: Andrew Chermak
published in: Journal of Algebra
publication date: 1998-11-22T00:00:00Z
review text: Let \(S\) be a finite nonempty set, and let \(M=(m_{a,b})_{a,b\in S}\) be a Coxeter matrix over \(S\). For \(m\) in \(\mathbb{N}\cup\{\infty\}\) and \(a,b\) in \(S\), let \(w(a,b;m)\) denote \((ab)^{{m\over 2}}\) if \(m\) is even, \((ab)^{{m-1\over 2}}a\) if \(m\) is odd, and \(1\) if \(m\) is \(\infty\). The Artin group associated with \(M\) is the group \(G(M)\) presented by \(S\) subject to the relations \(w(a,b;m_{a,b})=w(b,a;m_{a,b})\) for all \(a,b\) in \(S\). We say further that \(M\) and \(G(M)\) are locally non-spherical if \[ {1\over m_{a,b}}+{1\over m_{a,c}}+{1\over m_{b,c}}\leq 1\quad\text{for all } a,b,c\text{ in }S,\;a\neq b\neq c\neq a. \] Consider a locally non-spherical Artin group \(G\). The author exhibits an explicit solution of the word problem for \(G\) based on the study of a certain set of hyperbolic isometries of a tree. Consider a nonempty subset \(T\) of \(S\), let \(M_T\) be \((m_{a,b})_{a,b\in T}\), and let \(G_T\) be the subgroup of \(G\) generated by \(T\). The author also proves that \(G_T\) is naturally isomorphic to the Artin group associated with \(M_T\).
Mathematics Subject Classification ID: 20F36
Mathematics Subject Classification ID: 20F10
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 20E08
zbMATH DE Number: 1139769
zbMATH Keywords: Coxeter matrices
zbMATH Keywords: relations
zbMATH Keywords: locally non-spherical Artin groups
zbMATH Keywords: word problem
zbMATH Keywords: hyperbolic isometries
zbMATH Keywords: trees
reviewed by: Luis Paris
MaRDI profile type: Publication
full work available at URL: jabr.1997.7229
OpenAlex ID: W2050557236
cites work: Artin groups and infinite Coxeter groups
cites work: Artin groups of finite type are biautomatic
cites work: A convergence theorem for \(\Lambda\)-trees
cites work: Q3341041
cites work: Q5617560
DOI: 10.1006/JABR.1997.7229
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description: scientific article; zbMATH DE number 1139769