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Residual torsion-free nilpotence, bi-orderability, and two-bridge links

Part of: Ordered structures Low-dimensional topology

Published online by Cambridge University Press:  25 January 2023

Jonathan Johnson Open the ORCID record for Jonathan Johnson [Opens in a new window]
Jonathan Johnson*
Affiliation:
Department of Mathematics, Oklahoma State University, Stillwater, OK, USA
*
e-mail: jonathan.x.johnson@gmail.com
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Abstract

Residual torsion-free nilpotence has proved to be an important property for knot groups with applications to bi-orderability and ribbon concordance. Mayland proposed a strategy to show that a two-bridge knot group has a commutator subgroup which is a union of an ascending chain of para-free groups. This paper proves Mayland’s assertion and expands the result to the subgroups of two-bridge link groups that correspond to the kernels of maps to $\mathbb{Z}$. We call these kernels the Alexander subgroups of the links. As a result, we show the bi-orderability of a large family of two-bridge link groups. This proof makes use of a modified version of a graph-theoretic construction of Hirasawa and Murasugi in order to understand the structure of the Alexander subgroup for a two-bridge link group.

Keywords

Two-bridge knotsresidual nilpotencebi-orderable groupsAlexander subgroups

MSC classification

Primary: 57M05: Fundamental group, presentations, free differential calculus
Secondary: 06F15: Ordered groups
Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 64
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

The work was supported by NSF grants DMS-1937215 and DMS-2213213.

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