On graphic arrangement groups

TL;DR

This paper studies the algebraic and topological properties of groups derived from graphs, showing how certain graph conditions lead to embeddings into free groups and analyzing their finiteness properties.

Contribution

It introduces new embedding results for graphic arrangement groups based on graph structure and extends homological finiteness results to related groups.

Findings

Embedding of $P_ ext{ extGamma}$ into free groups for $K_4$-free graphs

Residually free and torsion-free properties of these groups

Finiteness type results depending on the number of $K_3$ subgraphs

Abstract

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.