Torsion in the quotient group of the Artin-Brieskorn braid group and regular Springer numbers
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 \(n\).
generalized Artin-Brieskorn braid groups
elements of finite order in braid groups
scientific article; zbMATH DE number 1002978