Fundamental groups of finite volume, bounded negatively curved 4-manifolds are not 3-manifold groups

We study noncompact, complete, finite volume, Riemannian 4-manifolds $M$ with sectional curvature $-1<K<0$. We prove that ${\pi }_{1}M$ cannot be a 3-manifold group. A classical theorem of Gromov says that $M$ is homeomorphic to the interior of a compact manifold $\text{M}$ with boundary $\mathrm{\partial }\text{barM}$. We show that for each ${\pi }_1$-injective boundary component $C$ of $\text{M}$, the map $i_{\ast }$ induced by inclusion $i\text{\hspace{0.17em}⁣}:C\to \text{M}$ has infinite index image $i_{\ast }({\pi }_{1}C)$ in ${\pi }_1\text{M}$. We also prove that $M$ cannot be homotoped to be contained in $\mathrm{\partial }\text{M}$.