Actions of the Braid Group, and New Algebraic Proofs of Results of Dehornoy and Larue

This article surveys many standard results about the braid group with emphasis on simplifying the usual algebraic proofs. We use van der Waerden's trick to illuminate the Artin-Magnus proof of the classic presentation of the algebraic mapping-class group of a punctured disc. We give a simple, new proof of the Dehornoy-Larue braid-group trichotomy, and, hence, recover the Dehornoy right-ordering of the braid group. We give three proofs of the Birman-Hilden theorem concerning the fidelity of braid-group actions on free products of finite cyclic groups, and discuss the consequences derived by Perron-Vannier, and the connections with Artin group (R) and the Wada representations. The first, very direct, proof, is due to Crisp-Paris and uses the sigma(1)-trichotomy and the Larue Shpilrain technique. The second proof arises by studying ends of free groups, and gives interesting extra information. The third proof arises from Larue's study of polygonal curves in discs with punctures, and gives extremely detailed information.