Appendix 1 - A presentation for the abstract coloured braid group (by Lars Gæde)

Summary

The purpose of this appendix is to derive the presentation stated in Lemma 1.4.2, p. 20, of the abstract coloured braid group, H_n, which geometrically corresponds to those braids that leave the order of the strings unchanged. There appears to be slight mistakes in several of the presentations found in the literature, and hence it seems appropriate to give a detailed derivation, so as to make it possible for the reader to check our computations.

Our main tool will be a general method of finding presentations of subgroups, called the Reidemeister – Schreier Rewriting Process. We apply this – not to H_n directly, since the computations involved are rather messy – but instead to the group D_n of braids that, geometrically speaking, do not change the position of the n'th string. The presentation obtained shows that D_n is isomorphic to a semi-direct product of the abstract braid group on n – 1 strings and a free group, which in turn implies that H_n is isomorphic to a semi-direct product of H_n-1 and a free group. Using this as the basis of an induction argument finally yields the desired presentation of H_n.

References for the material in this appendix are:

W.L. Chow, “On the algebraic braid group”, Ann. of Math. 49(1948), 654–358.

D.L. Johnson, “Topics in the theory of group presentations”, Cambridge University Press, 1980.

H. Zieschang, E. Vogt, H.–D. Coldewey, “Surfaces and planar discontinuous groups”, Springer–Verlag, 1980.