Juan Gonzalez-Meneses, Bert Wiest
Published 2003-05-11, updated 2003-09-12Version 2
The mixed braid groups are the subgroups of Artin braid groups whose elements preserve a given partition of the base points. We prove that the centralizer of any braid can be expressed in terms of semidirect and direct products of mixed braid groups. Then we construct a generating set of the centralizer of any braid on n strands, which has at most k(k+1)/2 elements if n=2k, and at most $k(k+3)/2 elements if n=2k+1. These bounds are shown to be sharp, due to work of N.V.Ivanov and of S.J.Lee. Finally, we describe how one can explicitly compute this generating set.
Comments: Section 5.3 is rewritten. The proposed generating set is shown not to be minimal, even though it is the smallest one reflecting the geometric approach. Proper credit is given to the work of other researchers, notably to N.V.Ivanov
Computation of Centralizers in Braid groups and Garside Groups
Faithful Linear Representations of the Braid Groups
Bieberbach groups and flat manifolds with finite abelian holonomy from Artin braid groups
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