Projective resolutions for graph products

Abstract

Let $\Gamma$ be a finite graph together with a group $G_v$ at each vertex $v$. The graph product $G(\Gamma )$ 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 $G(\Gamma )$ given projective resolutions for each $G_v$, and obtain some applications.

AMS-Tex, 4 pages, no figures