Graph of groups decompositions of graph braid groups

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.

arXiv commentVersion accepted for publication. Several minor corrections and clarifications have been made and new classes of examples have been added. Code is also available, implementing algorithms in this paper to compute free splittings and presentations of graph braid groups; see https://github.com/danberlyne/graph-braid-splitter and https://github.com/danberlyne/graph-braid-presenter

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