# Linearity of graph products

## TL;DR

This paper proves that graph products of linear groups over a domain are themselves linear over a polynomial extension, solving an open problem about their linearity over the complex numbers.

## Contribution

It establishes the linearity of graph products over polynomial rings, extending known results and resolving a specific open problem for faithful complex representations.

## Key findings

- Graph products of linear groups are linear over polynomial rings.
- Any graph product of finitely many complex linear groups is complex linear.
- The result solves an open problem by Hsu and Wise regarding faithful representations.

## Abstract

In this work we prove that, given a simplicial graph $\Gamma$ and a family $\mathcal{G}$ of linear groups over a domain $R$, the graph product $\Gamma\mathcal{G}$ 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 $\mathbb C$.