On graphic arrangement groups

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 K₄-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 K₃ 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.