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

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.

Minor changes, references added

Cited by in corpus (5)

• Generalized torsion elements and bi-orderability of 3-manifold groups
• Testing bi-orderability of knot groups
• Alexander polynomial obstruction of bi-orderability for rationally homologically fibered knot groups
• Residual Torsion-Free Nilpotence, Bi-Orderability and Pretzel Knots
• Generalized torsion and Dehn filling