Hyperplane complements of large type
type: Item
instance of: scholarly article
title: Hyperplane complements of large type
zbMATH Open document ID: 0564.57016
DOI: 10.1007/BF01388979
author: Harrie Hendriks
published in: Inventiones Mathematicae
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143201
review text: Let \(\Omega\) be a convex open subset of \(V={\mathbb{R}}^ n\), and \b{M} a collection of linear hyperplanes which is locally finite on \(\Omega\). Then \(\Omega\) is partitioned using \b{M} in facets. \b{M} is of large type on \(\Omega\) if each plinth P (i.e. codimension 2 facet) has multiplicity {\#}\(\{\) \(M\in \underline M\); \(P\subset M\}\geq 3\). We prove that if \b{M} is of large type on \(\Omega\), the complex hyperplane complement \(Y=\{z\in V\otimes {\mathbb{C}}\); Im(z)\(\in \Omega \}-\cup \{M\otimes {\mathbb{C}}\); \(M\in M\}\) is an Eilenberg-MacLane space of type K(\(\pi\),1). This is analogous to a theorem of \textit{P. Deligne} [ibid. 17, 273-302 (1972; Zbl 0238.20034)]. The proof is homotopy theoretical. The most striking consequence of the condition of large type is that no facets of codimension greater than 2 exist. The method allows us to prove the following ''Freiheitssatz'': Let G be an Artin group, i.e. G has finitely many generators \(s_ i\) and relations \(s_ is_ js_ i\cdot \cdot \cdot =s_ js_ is_ j\cdot \cdot \cdot\) \((m_{ij}\) factors on both sides, \(m_{ij}=\infty\) having no consequence). Suppose G is of large type, i.e. \(3\leq m_{ij}\leq \infty\) for all i,j. Then the set \(\{s^ 2_ i;i\}\) freely generates a free subgroup of G. The condition of large type is taken from \textit{K. I. Appel} and \textit{P. E. Schupp} [ibid. 72, 201-220 (1983; Zbl 0536.20019)]. In that paper small cancellation theory is used and (a special case of) the above ''Freiheitssatz'' is obtained.
Mathematics Subject Classification ID: 57N65
Mathematics Subject Classification ID: 55P20
Mathematics Subject Classification ID: 51F15
Mathematics Subject Classification ID: 32Sxx
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20F36
zbMATH DE Number: 3899768
zbMATH Keywords: complex hyperplane complements
zbMATH Keywords: Coxeter group
zbMATH Keywords: Eilenberg-MacLane space
zbMATH Keywords: Freiheitssatz
zbMATH Keywords: Artin group
MaRDI profile type: Publication
cites work: Artin groups and infinite Coxeter groups
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description: scientific article; zbMATH DE number 3899768