Mathematics – Group Theory
Scientific paper
2007-05-04
Groups Complexity Cryptology 1 (2009) 77-129.
Mathematics
Group Theory
51`pages, 13 figures
Scientific paper
10.1515/GCC.2009.77
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 then turn to the Birman-Hilden theorem concerning braid-group actions on free products of cyclic groups, and the consequences derived by Perron-Vannier, and the connections with the Wada representations. We recall the very simple Crisp-Paris proof of the Birman-Hilden theorem that uses the Larue-Shpilrain technique. Studying ends of free groups permits a deeper understanding of the braid group; this gives us a generalization of the Birman-Hilden theorem. Studying Jordan curves in the punctured disc permits a still deeper understanding of the braid group; this gave Larue, in his PhD thesis, correspondingly deeper results, and, in an appendix, we recall the essence of Larue's thesis, giving simpler combinatorial proofs.
Bacardit Lluis
Dicks Warren
No associations
LandOfFree
Actions of the braid group, and new algebraic proofs of results of Dehornoy and Larue does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Actions of the braid group, and new algebraic proofs of results of Dehornoy and Larue, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Actions of the braid group, and new algebraic proofs of results of Dehornoy and Larue will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-615210