Residual Torsion-Free Nilpotence, Bi-Orderability and Two-Bridge Links

TL;DR

This paper proves that certain subgroups of two-bridge link groups are unions of parafree groups, establishing bi-orderability for a large class of these groups using graph theory and algebraic techniques.

Contribution

It confirms Mayland's conjecture and extends the result to Alexander subgroups of two-bridge link groups, demonstrating their residual torsion-free nilpotence and bi-orderability.

Findings

Proved that Alexander subgroups form unions of parafree groups.

Established bi-orderability for a large family of two-bridge link groups.

Used graph theoretic methods to analyze subgroup structures.

Abstract

Residual torsion-free nilpotence has proven 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 parafree 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.