LEFT ORDERINGS AND QUOTIENTS OF THE BRAID GROUPS

Abstract

In this note, we show that a homomorphism from the braid group Bn into an arbitrary group is injective if and only if its image can be left ordered in a particular way. It follows from this criterion that the image of Bn (n ≥ 2) under any homomorphism is bi-orderable if and only if it is infinite cyclic. As a second application, this criterion also provides a new proof that the Artin representation of Bn into Aut (Fn) is injective.