Universal Structure of Graph Product Kernels

Let $G_{\mathrm{\Gamma }}$ be a graph product over a finite simplicial graph $\mathrm{\Gamma }$, and let $K_{\mathrm{\Gamma }}$ denote the kernel of the canonical homomorphism from $G_{\mathrm{\Gamma }}$ to the direct product of its vertex groups. It is known that, up to isomorphism, $K_{\mathrm{\Gamma }}$ depends only on the underlying graph $\mathrm{\Gamma }$ 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.