Universal Structure of Graph Product Kernels

Let $G_\Gamma$ be a graph product over a finite simplicial graph $\Gamma$, and let $K_\Gamma$ denote the kernel of the canonical homomorphism from $G_\Gamma$ to the direct product of its vertex groups. It is known that, up to isomorphism, $K_\Gamma$ depends only on the underlying graph $\Gamma$ and the cardinalities of the vertex groups. In this paper we establish a functorial refinement of this fact. We show that any collection of set maps between the vertex groups naturally induces a homomorphism between the corresponding kernels, and that this construction is functorial. Several applications are discussed.