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A partial order on the symmetric group and new $$K(\pi,1)$$’s for the braid groups. (English) Zbl 1011.20040
The author defines a certain partial order in the symmetric group $$S_n$$ using the Cayley graph structure with respect to the set of all transpositions. Then a certain group $$\Gamma_n$$ is introduced, which turns out to be isomorphic to the Artin braid group with $$n$$ strands, so as to be generated by the elements geodesically governed by the $$n$$-cycle $$(1,2,\dots,n)\in S_n$$ with explicit defining relations. A certain simplicial complex induced on the vertex set $$\Gamma_n$$ is shown to be contractible.

##### MSC:
 20F36 Braid groups; Artin groups 20F05 Generators, relations, and presentations of groups 20B30 Symmetric groups 57M07 Topological methods in group theory
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##### References:
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