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The Hurwitz Action and Braid Group Orderings
 Theory Appl. Categ
, 2001
"... In connection with the socalled Hurwitz action of homeomorphisms in ramified covers we define a groupoid, which we call a ramification groupoid of the 2sphere, constructed as a certain path groupoid of the universal ramified cover of the 2sphere with finitely many markedpoints. Our approach to ra ..."
Abstract

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In connection with the socalled Hurwitz action of homeomorphisms in ramified covers we define a groupoid, which we call a ramification groupoid of the 2sphere, constructed as a certain path groupoid of the universal ramified cover of the 2sphere with finitely many markedpoints. 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 orderinvariant action in the ramification groupoid of the sphere with n + 1 markedpoints. Leftinvariant 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 orderpreserving action.