A 3-manifold group which is not four dimensional linear

We give examples of closed orientable graph 3-manifolds having a fundamental group which is not a subgroup of GL(4,F)$\mathit{GL}(4,\mathbb{F})$ for any field F$\mathbb{F}$. This answers a question in the Kirby problem list from 1977 which is credited to the late William Thurston.