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

Abstract: 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 $\M$ with boundary $\partial\barM$. We show that for each $\pi_1$-injective boundary component $C$ of $\M$, the map $i_$ induced by inclusion $i\colon C\rightarrow \M$ has infinite index image $i_(\pi_1 C)$ in $\pi_1 \M$. We also prove that $M$ cannot be homotoped to be contained in $\partial\M$.