Lluís Bacardit and Warren Dicks,
Actions of the braid group, and new algebraic proofs of results of Dehornoy and Larue.
Groups -- Complexity -- Cryptology 1(2009), 77-129.

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.

November 3, 2008 version, 52 pages, available as
 

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    • Addenda (June 19, 2011)

    On page 100, before Corollary 7.6, add the following:

    Part (i) of the next result was previously shown in Lemma 3.4 and Theorem 4.4 of
    David Bessis, A dual braid monoid for the free group, J. Algebra 302 (2006), 55-69.
    The proofs given there are essentially the same as the proofs of Proposition C.4 and Theorem D.2 given
    below, which are extracted from Larue's PhD thesis.

    On page 101, after the last line, add the following:

    The fact that (8.1.1) and (8.1.2) are surjective follows from Theorem 5.8.2 of
    H. Zieschang, E. Vogt and H-D Coldewey: Surfaces and planar discontinuous groups.
    Translated from the German by John Stillwell. Lecture Notes in Mathematics 835, Springer, Berlin (1980).

    On page 111, after the last line, add the following to Historical Remarks 9.8:

        Makoto Sakuma, A note on Wada's group invariants of links, Proc. Japan Acad. 67 (1991), 176-177
    observed that the third Wada action of Bn on < x[1\uparrow n] |   > induces an action of Bn on
    < x[1\uparrow n] | x2[1\uparrow n] > which, when pre-composed with the inversion-of-the-generators
    automorphism of Bn, clearly agrees with the usual (Artin) action of   Bn on \Sigma0,1,n(2) (which is faithful
    by the Birman-Hilden Theorem). It follows from this observation that the third Wada action is faithful.
        Shpilrain, unaware of Sakuma's article, tacitly observed that the second Wada action of Bn on
    < x[1\uparrow n] |   > induces an action of Bn on < x[1\uparrow n] | x2[1\uparrow n] > that clearly agrees with
    the Artin action of Bn on \Sigma0,1,n(2), which implies that the second Wada action is faithful.
        Sakuma's article proves that the group invariants of links associated with the third Wada action agree
    with the group invariants of links found by Wada for the second Wada action.
        The Mathematical Reviewer of Wada's article, unaware of Sakuma's article, reported that results of
    A. J. Kelly, Groups from link diagrams, Ph.D. Thesis, Warwick Univ., Coventry, 1991
    imply exactly Sakuma's result.
        Sakuma's result was made particularly transparent fourteen years later when Crisp-Paris found that the
    second and third Wada actions agree up to a change of basis of < x[1\uparrow n] |   >.

    Up to Warren Dicks' publications.