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Actions of the braid group, and new algebraic proofs of results of Dehornoy and Larue. (English) Zbl 1195.20041
Summary: 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 braid group considered as the algebraic mapping-class group of a disc with punctures.
We give a simple, new proof of the $$\sigma_1$$-trichotomy for the braid group, 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 groups 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.

##### MSC:
 20F36 Braid groups; Artin groups 20E36 Automorphisms of infinite groups 20F05 Generators, relations, and presentations of groups 20E05 Free nonabelian groups 20F60 Ordered groups (group-theoretic aspects) 57M07 Topological methods in group theory
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