The lower central series of a fiber-type arrangement
type: Item
instance of: scholarly article
title: The lower central series of a fiber-type arrangement
zbMATH Open document ID: 0574.55010
DOI: 10.1007/BF01394780
published in: Inventiones Mathematicae
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143278
review text: An \(\ell\)-arrangement is a finite collection of hyperplanes in \({\mathbb{C}}^{\ell}\) (supposed here through the origin). The complement M of the union of the hyperplanes is a topological space with fundamental group \(\Pi_ 1(M)=G\). Let \(G=G_ 1\), \(G_{n+1}=[G_ n,G]\) for \(n\geq 1\) be the lower central series and \(\phi_ n=rank G_ n\). The author proves that for arrangements of ''fiber type'' the Poincaré series of M satisfies the relation \(P_ M(-t)=\prod^{\infty}_{j=1}(1-t^ j)^{\phi_ j}\). This result is a generalisation of a result of \textit{T. Kohno} [ibid. 82, 57-75 (1985; see the review above)].
Mathematics Subject Classification ID: 55P62
Mathematics Subject Classification ID: 32Q99
Mathematics Subject Classification ID: 20F36
Mathematics Subject Classification ID: 20J06
zbMATH DE Number: 3917256
zbMATH Keywords: complement of a union of complex hyperplanes
zbMATH Keywords: pure braid group
zbMATH Keywords: fundamental group
zbMATH Keywords: lower central series
zbMATH Keywords: Poincaré series
author: Michael Falk
author: Richard Randell
reviewed by: Jean-Claude Thomas
MaRDI profile type: Publication
cites work: Q3261851
cites work: Free arrangements of hyperplanes and supersolvable lattices
cites work: On the holonomy Lie algebra and the nilpotent completion of the fundamental group of the complement of hypersurfaces
cites work: Q5513066
cites work: Combinatorics and topology of complements of hyperplanes
cites work: Q3663320
cites work: Q4748256
description: scientific article; zbMATH DE number 3917256