Basic results on braid groups

González-Meneses, Juan

Annales mathématiques Blaise Pascal, Tome 18 (2011) no. 1, pp. 15-59.

Cet article contient les notes d’un course donné par l’auteur à l’Ecole Franco-Espagnole Tresses in Pau, qui a eu lieu à Pau (France) en Octobre 2009. Il s’agit essentiellement d’une introduction aux différents points des vue et techniques qui peuvent être utilisées pour montrer des résultats dans les groupes de tresses. En utilisant ces techniques on montre quelques résultats bien connus dans les groupes de tresses, à savoir l’exactitude de la presentation d’Artin, le fait que les groupes de tresses sont sans torsion, ou que son centre est engendré par le full twist. On rappelle quelques solutions des problèmes du mot et de la conjugaison, et aussi que les racines d’une tresse sont toutes conjuguées. On décrit aussi le centralisateur d’une tresse donnée. La plupart des preuves sont classiques, en utilisant de la terminologie moderne. J’ai choisi celles qui je trouve plus simples ou plus jolies.

These are Lecture Notes of a course given by the author at the French-Spanish School Tresses in Pau, held in Pau (France) in October 2009. It is basically an introduction to distinct approaches and techniques that can be used to show results in braid groups. Using these techniques we provide several proofs of well known results in braid groups, namely the correctness of Artin’s presentation, that the braid group is torsion free, or that its center is generated by the full twist. We also recall some solutions of the word and conjugacy problems, and that roots of a braid are always conjugate. We also describe the centralizer of a given braid. Most proofs are classical ones, using modern terminology. I have chosen those which I find simpler or more beautiful.

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