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The Hurwitz action and braid group orderings

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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.

Keywords:

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

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

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