Torsion in the quotient group of the Artin-Brieskorn braid group and regular Springer numbers
type: Item
instance of: scholarly article
title: Torsion in the quotient group of the Artin-Brieskorn braid group and regular Springer numbers
zbMATH Open document ID: 0878.20026
DOI: 10.1007/BF02509554
author: O. V. Shvartsman
publication date: 1997-11-24T00:00:00Z
review text: Let \(W\) be a finite Coxeter group acting on the complex vector space \(V\) and denote by \(Y\) the complement of the set of fixed points of elements of \(W\). Then the fundamental group of \(Y\) is the generalized Artin-Brieskorn braid group \(B(W)\) [\textit{E. Brieskorn}, Invent. Math. 12, 57-61 (1971; Zbl 0204.56502)]. We obtain an action of \(W\) on the projective space of \(V\) and a free action on the complement \(P\) of the set of fixed points of elements of \(W\) acting in a non trivial way. Let us assume that \(W\) is irreducible and maximal with respect to inclusion in the set of all Coxeter groups defining the same projective group \(G\). Then, the author shows that the fundamental group of \(P/G\) is isomorphic to \(B(W)\) modulo its center. Furthermore he determines the orders of torsion elements. As a corollary, one obtains for \(n\geq 5\) the Murasugi theorem [\textit{K. Murasugi}, Proc. Lond. Math. Soc., III. Ser. 44, 71-84 (1982; Zbl 0489.57003)] that the order of a torsion element in the quotient group \(B(n)/Z(B(n))\) of the Artin braid group of \(n\) strands has to divide \(n-1\) or \(n\).
Mathematics Subject Classification ID: 20F36
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 1002978
zbMATH Keywords: generalized Artin-Brieskorn braid groups
zbMATH Keywords: elements of finite order in braid groups
zbMATH Keywords: Springer numbers
zbMATH Keywords: finite Coxeter groups
zbMATH Keywords: fundamental groups
zbMATH Keywords: free actions
reviewed by: Ludwig Balke
MaRDI profile type: Publication
cites work: Q5580319
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published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 1002978