Residual (torsion-free nilpotence) in one-relator groups

Andrew Glass (Cambridge)

Frank Adams 1,

We seek necessary and sufficient conditions for a one-relator group to be bi-orderable.
These concern the roots of an Alexander-Conway polynomial and answer questions about certain knot groups.
The key tool is:

Theorem: Let \(G\) be an extension of a residually (torsion-free nilpotent) group \(K\) by a bi-orderable abelian group \(\Phi\).
Suppose that the real vector space \(V=K^{\rm ab}\otimes \mathbb{R}\) is finite-dimensional, and that all eigenvalues of maps induced on \(V\) by elements of \(\Phi\) are positive real numbers. Then \(G\) is bi-orderable.

We outline the proof and explain how it provides our conditions.

This is joint work with I. M. Chiswell and John S. Wilson (arXiv: 1405.0994).

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