Deux Propriétés Des Groupes De Tresses. Two Properties of Braid Groups

We describe a new algorithm for comparing braid words, and establish some order properties for the braid groups which are connected with left distributive operations. Abridged English Version. We apply to braids some tools used in the study of left distributive operations (c.f. [2]). The results deal with special decompositions of braid words. One describes a new quadratic algorithm for comparing braid words (c.f. [3]). Denote by W and W+ respectively the free monoids of braid words and of positive braid words. Theorem 1.There exist two mappings DLR and NLR of W into W+ such that i) for every α in W, α and DLR(α)−1NLR(α) represent the same braid ; ii) the word α represents the trivial braid if and only if the words DLR(α) and NLR(α) are empty ; iii) when only a fixed number of generators are used, the determination of DLR(α) and NLR(α) for α with length N requires O(N) steps. One also transfers to braids some ordering properties previously established for free distributive structures. Theorem 2.For any i, a braid word with at least one occurrence of the generator σi but no occurrence of σ −1 i cannot be trivial. Theorem 3.There exists a unique ordering on the braid group B∞ which is compatible with the product on the left and is such that every generator is infinitely large with respect to the family of all σk with k > i. This ordering is linear, and there exists an effective algorithm for comparing braid words. The author thanks S. Burckel, G. Duchamp, Ch. Kassel and P. Vogel for their interest in the subject.