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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Practically perfect three-manifold groups

Author: D. R. McMillan
Journal: Proc. Amer. Math. Soc. 49 (1975), 481-486
MSC: Primary 57A10
MathSciNet review: 0368011
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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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Keywords: Fundamental group, $ 3$-manifold, connected sum, free product, absolute neighborhood retract, aspherical, perfect group
Article copyright: © Copyright 1975 American Mathematical Society