Group rings of three-manifold groups

Let $G$ be the fundamental group of a three-manifold. By piecing together many known facts about three manifold groups, we establish two properties of the group ring $\mathbb{C}G$. We show that if $G$ has rational cohomological dimension two, then $\mathbb{C}G$ is coherent. We also show that if $G$ is torsion-free, then $G$ satisfies the Strong Atiyah Conjecture over $\mathbb{C}$ and hence that $\mathbb{C}G$ satisfies Kaplansky’s Zero Divisor Conjecture.