Linearity of graph products

Abstract

In this work we prove that, given a simplicial graph $Γ$ and a family $\mathcal{G}$ of linear groups over a domain $R$, the graph product $Γ\mathcal{G}$ is linear over $R[\underline t]$, where $\underline t$ is a tuple of finitely many linearly independent variables. As a consequence we obtain that any graph product of finitely many groups linear over the complex numbers is again a linear group over the complex numbers. This solves an open problem of Hsu and Wise in the case of faithful representations over $\mathbb C$.