Graph braid groups and right-angled Artin groups

TL;DR

This paper characterizes when a graph's braid group is a right-angled Artin group for braid index at least 5, linking graph planarity to algebraic properties of the braid group.

Contribution

It provides a necessary and sufficient condition for graphs to have right-angled Artin groups as their braid groups for braid index ≥ 5, and explores related conjectures.

Findings

Graph planarity iff first homology of 2-braid group is torsion-free

Classification of graphs with right-angled Artin braid groups for index ≥ 5

Formulation of conjectures for braid groups with index ≤ 4

Abstract

We give a necessary and sufficient condition for a graph to have a right-angled Artin group as its braid group for braid index $\ge 5$. In order to have the necessity part, graphs are organized into small classes so that one of homological or cohomological characteristics of right-angled Artin groups can be applied. Finally we show that a given graph is planar iff the first homology of its 2-braid group is torsion-free and leave the corresponding statement for $n$-braid groups as a conjecture along with few other conjectures about graphs whose braid groups of index $\le 4$ are right-angled Artin groups.