## Abstract

A finite simple graph Γ determines a quotient P_{Γ} of the pure braid group, called a graphic arrangement group. We analyze homomorphisms of these groups defined by deletion of sets of vertices, using methods developed in prior joint work with R. Randell. We show that, for a K_{4}-free graph Γ, a product of deletion maps is injective, embedding P_{Γ} in a product of free groups. Then P_{Γ} is residually free, torsion-free, residually torsion-free nilpotent, and acts properly on a CAT(0) cube complex. We also show P_{Γ} is of homological finiteness type F_{m−1}, but not F_{m}, where m is the number of copies of K_{3} in Γ, except in trivial cases. The embedding result is extended to graphs whose 4-cliques share at most one edge, giving an injection of P_{Γ} into the product of pure braid groups corresponding to maximal cliques of Γ. We give examples showing that this map may inject in more general circumstances. We define the graphic braid group B_{Γ} as a natural extension of P_{Γ} by the automorphism group of Γ, and extend our homological finiteness result to these groups.

Original language | English (US) |
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Article number | 102130 |

Journal | Advances in Applied Mathematics |

Volume | 126 |

DOIs | |

State | Published - May 2021 |

## Keywords

- Graphic arrangement
- Homological finiteness type
- Hyperplane arrangement
- K-free graph
- Pure braid group

## ASJC Scopus subject areas

- Applied Mathematics