Graph 4-braid groups and Massey products

TL;DR

This paper investigates the algebraic structure of 4-braid groups over graphs, establishing conditions under which these groups are right-angled Artin groups based on the graph's topology and Massey products.

Contribution

It characterizes when 4-braid groups are right-angled Artin groups using topological and algebraic tools, linking graph topology with braid group properties.

Findings

Graphs without certain subgraphs have braid groups with commutator-only presentations.

The absence of four specific subgraphs characterizes when 4-braid groups are right-angled Artin groups.

Discrete Morse theory and Massey products are key tools in the analysis.

Abstract

We first show that the braid group over a graph topologically containing no $\Theta$-shape subgraph has a presentation related only by commutators. Then using discrete Morse theory and triple Massey products, we prove that a graph topologically contains none of four prescribed graphs if and only if its 4-braid groups is a right-angled Artin group.