Residual nilpotence and ordering in one-relator groups and knot groups

TL;DR

This paper investigates the bi-orderability of one-relator groups, especially knot groups, by linking algebraic properties of associated polynomials to orderability, extending Baumslag's work on group roots.

Contribution

It establishes conditions relating roots of Alexander polynomials to bi-orderability of certain groups, extending existing theories and addressing open questions.

Findings

Bi-orderability linked to roots of Alexander polynomial

All roots real and positive imply bi-orderability

At least one root real and positive if bi-orderable

Abstract

Let $G=< x,t\mid w>$ be a one-relator group, where $w$ is a word in $x,t$. If $w$ is a product of conjugates of $x$ then, associated with $w$, there is a polynomial $A_w(X)$ over the integers, which in the case when $G$ is a knot group, is the Alexander polynomial of the knot. We prove, subject to certain restrictions on $w$, that if all roots of $A_w(X)$ are real and positive then $G$ is bi-orderable, and that if $G$ is bi-orderable then at least one root is real and positive. This sheds light on the bi-orderability of certain knot groups and on a question of Clay and Rolfsen. One of the results relies on an extension of work of G. Baumslag on adjunction of roots to groups, and this may have independent interest.