On graphic arrangement groups

Abstract

A finite simple graph Γ determines a quotient PΓ 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 PΓ in a product of free groups. Then PΓ is residually free, torsion-free, residually torsion-free nilpotent, and acts properly on a CAT(0) cube complex. We also show PΓ is of homological finiteness type Fm−1, but not Fm, 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 PΓ 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 BΓ as a natural extension of PΓ by the automorphism group of Γ, and extend our homological finiteness result to these groups.