Projective resolutions for graph products

Abstract

Let $Γ$ be a finite graph together with a group $G_v$ at each vertex $v$. The graph product $G(Γ)$ is obtained from the free product of all $G_v$ by factoring out by the normal subgroup generated by $\{g^{-1}h^{-1}gh; g\in G_v, h\in G_w\}$ for all adjacent $v,w$. In this note we construct a projective resolution for $G(Γ)$ given projective resolutions for each $G_v$, and obtain some applications.