On Tit's conjecture and other questions concerning Artin and generalized Artin groups
type: Item
instance of: scholarly article
title: On Tit's conjecture and other questions concerning Artin and generalized Artin groups
zbMATH Open document ID: 0633.20021
DOI: 10.1007/BF01389074
author: Stephen J. Pride
published in: Inventiones Mathematicae
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143398
review text: Let \(\Gamma\) be a graph every edge in which is determined by two vertices x, y from the vertex set X not excepting \(x=y\). Suppose that there is a function \(\phi\) which assigns to each edge \(\{\) x,y\(\}\) of \(\Gamma\) a word of the form \((xyx...)(yxy...)^{-1}\), where the two bracketed words have the same length \(>1\). The group G(\(\Gamma\),\(\phi)\) with the presentation \(<X\|\phi\{\) x,y\(\}\), (\(\{\) x,y\(\}\) an edge in \(\Gamma)>\) is called an Artin group. More generally let the function \(\phi\) assign to each edge \(\{\) x,y\(\}\) of \(\Gamma\) a word of the form \(x^{\epsilon_ 1}y^{\delta_ 1}x^{\epsilon_ 2}y^{\delta_ 2}...x^{\epsilon_ n}y^{\delta_ n}\) where \(n\geq 2\), \(| \epsilon_ i| =| \delta_ i| =1\) for \(i=1,2,...,n\). The group \(G=G(\Gamma,\phi)\) defined as above is called a generalized Artin group. The author gives necessary and sufficient conditions for the group \(G=G(\Gamma,\phi)\) to satisfy the small cancellation conditions C(4) and T(4). If it is so then the finitely presented group G(\(\Gamma\),\(\phi)\) has solvable word and conjugacy problems. Also every relation between the elements \(x^ 2\) (x\(\in X)\) is a consequence of the relations \(y^ 2z^ 2=z^ 2y^ 2\) where \(\{\) y,z\(\}\) ranges over all edges of \(\Gamma\) for which \(\phi\{\) y,z\(\}\) is conjugated to one of the words \([y^{\epsilon},z^{\delta}]\), \(| \epsilon | =| \delta | =1\), or \((st)^ 2(st^{-1})^ 2\).
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20F06
Mathematics Subject Classification ID: 20F10
Mathematics Subject Classification ID: 20F36
Mathematics Subject Classification ID: 05C25
zbMATH DE Number: 4029822
zbMATH Keywords: words
zbMATH Keywords: presentation
zbMATH Keywords: generalized Artin group
zbMATH Keywords: small cancellation conditions
zbMATH Keywords: finitely presented group
zbMATH Keywords: solvable word and conjugacy problems
Wikidata QID: Q122908240
reviewed by: Vitaliĭ Roman'kov
MaRDI profile type: Publication
cites work: Q3697232
cites work: Artin groups and infinite Coxeter groups
cites work: Kommutationsgleichungen in semifreien gruppen
cites work: Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe
cites work: Artin-Gruppen und Coxeter-Gruppen
cites work: Les immeubles des groupes de tresses généralises
cites work: Hyperplane complements of large type
cites work: Q4145882
cites work: On generalized braid groups
cites work: On the T(q)-conditions of small cancellation theory
description: scientific article; zbMATH DE number 4029822