Basic results on braid groups

TL;DR

This paper provides an accessible introduction to braid groups, covering classical results and techniques with modern terminology, including proofs of Artin's presentation, torsion-freeness, and the structure of the center.

Contribution

It offers a clear, modern presentation of fundamental results in braid groups, emphasizing simpler and more elegant proofs for key properties.

Findings

Artin's presentation is correct

Braid groups are torsion-free

The center is generated by the full twist

Abstract

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.