Graph of groups decompositions of graph braid groups

TL;DR

This paper introduces a method to decompose graph braid groups into a graph of groups, enabling easier computation and analysis of their structure, including criteria for splitting and distinctions among related groups.

Contribution

It provides an explicit construction for decomposing graph braid groups, answers key questions about their splitting properties, and distinguishes certain right-angled Artin groups from graph braid groups.

Findings

Decomposition method for graph braid groups as a graph of groups

Criteria for when a graph braid group splits as a free product

An example of a relatively hyperbolic graph braid group not hyperbolic relative to subgroups

Abstract

We provide an explicit construction that allows one to easily decompose a graph braid group as a graph of groups. This allows us to compute the braid groups of a wide range of graphs, as well as providing two general criteria for a graph braid group to split as a non-trivial free product, answering two questions of Genevois. We also use this to distinguish certain right-angled Artin groups and graph braid groups. Additionally, we provide an explicit example of a graph braid group that is relatively hyperbolic, but is not hyperbolic relative to braid groups of proper subgraphs. This answers another question of Genevois in the negative.