Subgroup separability, knot groups and graph manifolds

Niblo, Graham; Wise, Daniel

doi:10.1090/S0002-9939-00-05574-X

Subgroup separability, knot groups and graph manifolds
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by Graham A. Niblo and Daniel T. Wise

Proc. Amer. Math. Soc. 129 (2001), 685-693

DOI: https://doi.org/10.1090/S0002-9939-00-05574-X

Published electronically: April 27, 2000

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Abstract:

This paper answers a question of Burns, Karrass and Solitar by giving examples of knot and link groups which are not subgroup-separable. For instance, it is shown that the fundamental group of the square knot complement is not subgroup separable. Let $L$ denote the fundamental group of the link consisting of a chain of $4$ circles. It is shown that $L$ is not subgroup separable. Furthermore, it is shown that $L$ is a subgroup of every known non-subgroup separable compact 3-manifold group. It is asked whether all such examples contain $L$.

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