Examples of centralizers in the Artin braid groups

The normalizer of an element of a group is defined as the subgroup of all elements commuting with it; this subgroup is often also called the centralizer of the given element. In the second version [F-GM] the authors switched to the latter terminology, and we will also use the term “centralizer”. The conjecture was supported by a new algorithm for finding generators of a centralizer, suggested in [F-GM], and by extensive computations based on this algorithm. The goal of this paper is to present simple examples of elements of Bn for which the centralizer cannot be generated by a less than quadratic in n number of generators. In particular, the above conjecture is disproved. The key insight is that the centralizer may be closer to a pure braid group than to a braid group (as one may think initially), and that a pure braid group requires the number of generators quadratic in the number of strings. Our methods are based on Thurston’s theory of the surface diffeomorphisms and on the well known relation between the Artin braid groups and mapping class groups, in contrast with the purely algebraic methods of [F-GM]. In fact, the effectiveness of Thurston’s theory for studying the centralizers in the context of the mapping class groups was established long