Practically Perfect Three-Manifold Groups

Abstract

Let ${M^3}$ be a $3$-manifold containing no $2$-sided projective plane. Let $G \ne \{ 1\}$ be a finitely-generated subgroup of ${\pi _1}({M^3})$ such that $G$ is indecomposable relative to free product, and such that $G$ abelianized is finite. ($G$ is “practically perfect".) Then, it is shown that there is a compact $3$-submanifold ${Z^3} \subset {M^3}$ such that ${\pi _1}({Z^3})$ contains a subgroup of finite index conjugate to $G$, and ${Z^3}$ is bounded by a $2$-sphere. Some related extensions of this result are given, plus an application to compact absolute neighborhood retracts in $3$-manifolds.