The Hurwitz action and braid group orderings (Q2762221)

This paper concerns the (left-invariant) linear ordering of the Artin braid group of \(n\) strands, as defined by \textit{P. Dehornoy} [``Braids and self-distributivity'', Prog. Math. 192 (2000; Zbl 0958.20033)], in particular its geometric interpretations. NEWLINENEWLINENEWLINEThe author is led to investigate the above through certain path groupoids of the universal cover of the 2-sphere with finitely many marked-points, and which he calls ``ramification groupoid''. It is defined in connection with the Hurwitz action of homeomorphisms on ramified covers, which are approached via cosheaf spaces or complete spreads. NEWLINENEWLINENEWLINEThe main result is that the ramification groupoid carries a certain order structure and that the Artin group of braids on \(n\)-strands has an order-invariant action on the ramification groupoid of the sphere with \(n+1\)-marked points. Also, it is shown here that the left-invariant linear orderings of the braid groups, including the Dehornoy ordering, can be retrieved. NEWLINENEWLINENEWLINEThere is a tangential use of topos theory which ought not to detract the reader not acquainted with it.