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A ribbon knot group which has no free base


Author: Katsuyuki Yoshikawa
Journal: Proc. Amer. Math. Soc. 102 (1988), 1065-1070
MSC: Primary 57Q45; Secondary 20E06, 20F05
DOI: https://doi.org/10.1090/S0002-9939-1988-0934891-7
MathSciNet review: 934891
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Abstract: We consider the following problem: If a group $ G$ satisfies the conditions (1) $ G$ has a finite presentation with $ r + 1$ generators and $ r$ relators, and (2) there exists an element $ x$ of $ G$ such that $ G = {\left\langle {\left\langle x \right\rangle } \right\rangle ^G}$ where $ {\left\langle {\left\langle x \right\rangle } \right\rangle ^G}$ is the normal closure of $ x$ in $ G$, then is $ G$ an HNN (Higman-Neumann-Neumann) extension of a free group of finite rank? In this paper, we give a negative answer to the problem. Thus it follows that there exists a ribbon $ n$-knot group $ (n \geq 2)$ which has no free base.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0934891-7
Keywords: Knot group, HNN extension
Article copyright: © Copyright 1988 American Mathematical Society

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