An Ordering for Groups of Pure Braids and Fibre-Type Hyperplane Arrangements

[摘要] We define a total ordering of the pure braid groups which isinvariant under multiplication on both sides. This ordering isnatural in several respects. Moreover, it well-orders the pure braidswhich are positive in the sense of Garside. The ordering is definedusing a combination of Artin's combing technique and the Magnusexpansion of free groups, and is explicit and algorithmic.By contrast, the full braid groups (on 3 or more strings) can beordered in such a way as to be invariant on one side or the other, butnot both simultaneously. Finally, we remark that the same type ofordering can be applied to the fundamental groups of certain complexhyperplane arrangements, a direct generalization of the pure braidgroups.

[学科分类] 数学（综合）