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A combinatorial genesis of the right-angled relations in Artin's classical braid groups

Omar Alvarado-Garduño, Jesús González, Matthew Kahle

TL;DR

The paper demonstrates that the fundamental groups of unlabelled configuration spaces in width-2 geometries, specifically UC(n,2) and UC(n,p×2), realize right-angled Artin groups that distill the RA relations of Artin's braid groups B_n while omitting the AT relations in these regimes. By applying Forman's discrete Morse theory and the Farley-Sabalka gradient on Abrams models, the authors build explicit Morse presentations and reduce them to minimal generators. For p≤n≤2p−5, they show these groups are RAAGs with commuting relations corresponding to nonconsecutive indices, establishing an algebraic genesis of the RA relations from geometric configuration spaces. This provides a precise geometric/combinatorial lens on the RA structure underlying B_n and outlines a pathway to extending RAAG realizations to more general width configurations.

Abstract

We show that the fundamental group of unlabelled configuration spaces of thick particles in either a width-2 infinite strip or a width-2 rectangle are right-angled Artin groups capturing the right-angled essence of Artin's braid groups.

A combinatorial genesis of the right-angled relations in Artin's classical braid groups

TL;DR

The paper demonstrates that the fundamental groups of unlabelled configuration spaces in width-2 geometries, specifically UC(n,2) and UC(n,p×2), realize right-angled Artin groups that distill the RA relations of Artin's braid groups B_n while omitting the AT relations in these regimes. By applying Forman's discrete Morse theory and the Farley-Sabalka gradient on Abrams models, the authors build explicit Morse presentations and reduce them to minimal generators. For p≤n≤2p−5, they show these groups are RAAGs with commuting relations corresponding to nonconsecutive indices, establishing an algebraic genesis of the RA relations from geometric configuration spaces. This provides a precise geometric/combinatorial lens on the RA structure underlying B_n and outlines a pathway to extending RAAG realizations to more general width configurations.

Abstract

We show that the fundamental group of unlabelled configuration spaces of thick particles in either a width-2 infinite strip or a width-2 rectangle are right-angled Artin groups capturing the right-angled essence of Artin's braid groups.

Paper Structure

This paper contains 10 sections, 21 theorems, 81 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Discrete Morse-Poincaré approach to the fundamental group
  4. Farley-Sabalka gradient field on Abrams model
  5. Discrete homotopy models
  6. Disks on an infinite strip
  7. Squares in a rectangle
  8. Raw presentation
  9. Minimal generators and key relations
  10. Proof conclusion

Key Result

Theorem 1.1

All groups $B_n(2)$ with $n\geq2$, and groups $B_n(p\times 2)$ with $n,p\geq2$ and either $n\leq p$ or $n\leq 2p-5$ are RAAGs generated by $n-1$ elements subject only to the RA-type relations in (rarelations).

Figures (9)

  • Figure 1: The simple collapse for a $W$-pair and the corresponding step in the reduction process
  • Figure 2: Grid $\Gamma_{6,4}$ (left) and its maximal tree (right)
  • Figure 3: A typical step in the construction of reduced forms
  • Figure 4: $W_{p,q,n}$-critical 2-cell in Case 1 (left) and the resulting relation (right)
  • Figure 5: $W_{p,q,n}$-critical 2-cell in Case 2 (left) and the resulting relation (right)
  • ...and 4 more figures

Theorems & Definitions (42)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 2.1
  • Theorem 2.2: Abrams, 2000
  • proof : Proof of Theorem \ref{['combgen']} in the case of $B_n(2)$
  • Proposition 3.1
  • ...and 32 more