Practically perfect three-manifold groups

McMillan, D. R.

doi:10.1090/S0002-9939-1975-0368011-8

Practically perfect three-manifold groups
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Proc. Amer. Math. Soc. 49 (1975), 481-486 Request permission

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.

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