[Paper Review] A class of Garside groupoid structures on the pure braid group

Daan Krammer|ArXiv.org|Sep 7, 2005

Geometric and Algebraic Topology9 references4 citations

TL;DR

This paper introduces a new class of Garside groupoid structures on the pure braid group, parameterized by labelings of punctures with integers ≥2. By constructing a groupoid acting on ball decompositions of a punctured disk—where labels correspond to region perimeters—it generalizes Garside's original structure and extends the Tamari lattice ordering to new finite lattices, including associahedra and permutahedra, depending on the labeling function.

ABSTRACT

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.

Motivation & Objective

• To generalize Garside's original structure on the braid group to a broader class of groupoid structures on the pure braid group.
• To define a new family of Garside groupoids indexed by functions from punctures to integers ≥2.
• To extend the Tamari lattice ordering to new finite lattices arising from ball decompositions of the punctured disk.
• To demonstrate that the interval [x, x∆] in the lattice forms a finite polytope, including associahedra and permutahedra under specific labelings.
• To provide a unified framework for understanding lattice orderings on triangulations of the n-gon via the groupoid action.

Proposed method

• Construct a braid-like groupoid acting on the set of ball decompositions of a punctured disk, where each region contains one puncture and its perimeter matches the puncture's label.
• Define a presentation of the groupoid using elementary relations (ER1–ER4) that encode local moves on decompositions.
• Establish that the groupoid satisfies the axioms of a Garside groupoid, including the existence of a Garside element ∆ and finite intervals [x, x∆].
• Use the action of the groupoid on a lattice of decompositions to define the interval [x, x∆] as a finite lattice, generalizing the Tamari lattice.
• Prove that when all labels are 3, the interval Ω(y) is in bijection with the set of triangulations of the n-gon, inducing a lattice ordering on this set.

Experimental results

Research questions

• RQ1 Can Garside groupoid structures be generalized beyond the classical braid group and Birman–Ko–Lee constructions?
• RQ2 How do labelings of punctures with integers ≥2 affect the structure of the pure braid group and its associated lattices?
• RQ3 What is the relationship between the interval [x, x∆] in the lattice and known polytopes such as the associahedron and permutahedron?
• RQ4 Under what conditions does the groupoid action yield a lattice ordering on the set of triangulations of a disk with labeled punctures?
• RQ5 Is the Tamari lattice ordering on triangulations a special case of a broader family of lattice orderings arising from Garside groupoids?

Key findings

• The paper constructs a new class of Garside groupoid structures on the pure braid group, one for each labeling function from punctures to integers ≥2.
• When all labels are 3, the interval [x, x∆] is in bijection with the set of triangulations of the n-gon, generalizing the Tamari lattice ordering.
• The structure generalizes Garside’s original Garside structure but does not include the Birman–Ko–Lee structure.
• For all labels equal to 2, the groupoid action is transitive, resulting in a single lattice ordering on the permutahedron.
• The interval [x, x∆] forms a finite lattice, and when all labels are 3, this lattice is isomorphic to the Tamari lattice on the associahedron.
• The paper proves that the lattice ordering on [x, x∆] is induced by the groupoid action and is preserved under the Garside element ∆.