A finite simple graph $\Gamma$ determines a quotient $P_\Gamma$ 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 $\Gamma$, a product of deletion maps is injective, embedding $P_\Gamma$ in a product of free groups. Then $P_\Gamma$ is residually free, torsion-free, residually torsion-free nilpotent, and acts properly on a CAT(0) cube complex. We also show $P_\Gamma$ is of homological finiteness type $F_{m-1}$, but not $F_m$, where $m$ is the number of copies of $K_3$ in $\Gamma$, except in trivial cases. The embedding result is extended to graphs whose 4-cliques share at most one edge, giving an injection of $P_\Gamma$ into the product of pure braid groups corresponding to maximal cliques of $\Gamma$. We give examples showing that this map may inject in more general circumstances. We define the graphic braid group $B_\Gamma$ as a natural extension of $P_\Gamma$ by the automorphism group of $\Gamma$, and extend our homological finiteness result to these groups.

中文翻译：

---

关于图形排列组

一个有限简单图$\Gamma$决定了纯编织群的商$P_\Gamma$，称为图形排列群。我们使用与 R. Randell 先前联合工作中开发的方法，分析通过删除顶点集定义的这些组的同态性。我们证明，对于 $K_4$-free 图 $\Gamma$，删除映射的乘积是单射的，将 $P_\Gamma$ 嵌入到自由群的乘积中。那么 $P_\Gamma$ 是残余自由、无扭转、残余无扭转的幂零，并且在 CAT(0) 立方体复形上正确作用。我们还表明 $P_\Gamma$ 是同调有限类型 $F_{m-1}$，但不是 $F_m$，其中 $m$ 是 $\Gamma$ 中 $K_3$ 的副本数，除了平凡案件。嵌入结果扩展到 4-cliques 最多共享一条边的图，将 $P_\Gamma$ 注入与 $\Gamma$ 的最大团对应的纯编织群的乘积中。我们举例说明该映射可能会在更一般的情况下注入。我们将图形编织群$B_\Gamma$定义为$\Gamma$的自同构群对$P_\Gamma$的自然扩展，并将我们的同调有限性结果扩展到这些群。