标题： 辫群的作用，以及Dehornoy和Larue结果的新代数证明 显示英文标题

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

摘要： 本文综述了许多关于辫群的标准结果，并着重于简化通常的代数证明。 我们使用范德瓦尔登技巧来阐明阿廷-马格努斯对带有孔的圆盘的代数映射类群的经典表示的证明。 我们给出了德霍诺伊-拉鲁辫群三分法的新简单证明，从而恢复了辫群的德霍诺伊右序。 然后我们转向贝兰-希登定理，该定理涉及辫群在循环群自由积上的作用，以及佩龙-瓦尼尔得出的结论，以及与瓦达表示的联系。 我们回忆了克里普斯-帕里斯对贝兰-希登定理的非常简单的证明，该证明使用了拉鲁-什皮莱因技术。 研究自由群的末端有助于更深入地理解辫群；这使我们能够推广贝兰-希登定理。 研究带有孔的圆盘中的若尔当曲线可以更深入地理解辫群；这使得拉鲁在他的博士论文中得到了相应更深入的结果，在附录中，我们回顾了拉鲁论文的核心内容，并给出了更简单的组合证明。 显示英文摘要

摘要： 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.