The Hurwitz action and braid group orderings

Jonathon Funk

In connection with the so-called Hurwitz action of homeomorphisms in ramified covers we define a groupoid, which we call a ramification groupoid of the 2-sphere, constructed as a certain path groupoid of the universal ramified cover of the 2-sphere with finitely many marked-points. Our approach to ramified covers is based on cosheaf spaces, which are closely related to Fox's complete spreads. A feature of a ramification groupoid is that it carries a certain order structure. The Artin group of braids of $n$ strands has an order-invariant action in the ramification groupoid of the sphere with $n+1$ marked-points. Left-invariant linear orderings of the braid group such as the Dehornoy ordering may be retrieved. Our work extends naturally to the braid group on countably many generators. In particular, we show that the underlying set of a free group on countably many generators (minus the identity element) can be linearly ordered in such a way that the classical Artin representation of a braid as an automorphism of the free group is an order-preserving action.


2000 MSC: 18D10, 18D30, 20F36.

Theory and Applications of Categories, Vol. 9, 2001, No. 7, pp 121-150.

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