Actions of the braid group, and new algebraic proofs of results of Dehornoy and Larue

TL;DR

This paper reviews key results about braid groups, offering simplified algebraic proofs, new insights into braid group orderings, and generalizations of classical theorems using innovative techniques and deeper structural analysis.

Contribution

It provides simplified proofs of fundamental braid group results, introduces a new proof of the Dehornoy-Larue trichotomy, and generalizes the Birman-Hilden theorem through novel approaches.

Findings

Simplified algebraic proofs of braid group properties

A new proof of the Dehornoy-Larue braid-group trichotomy

Generalization of the Birman-Hilden theorem

Abstract

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.