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A class of Garside groupoid structures on the pure braid group. (English) Zbl 1194.20040
Summary: We construct a class of Garside groupoid structures on the pure braid groups, one for each function (called labelling) from the punctures to the integers greater than 1. The object set of the groupoid is the set of ball decompositions of the punctured disk; the labels are the perimeters of the regions. Our construction generalises Garside’s original Garside structure, but not the one by Birman-Ko-Lee. As a consequence, we generalise the Tamari lattice ordering on the set of vertices of the associahedron.

MSC:
20F36 Braid groups; Artin groups
20F05 Generators, relations, and presentations of groups
20M05 Free semigroups, generators and relations, word problems
06A06 Partial orders, general
20F60 Ordered groups (group-theoretic aspects)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M07 Topological methods in group theory
20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms)
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References:
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