Department of Mathematics

A generalized torsion element of a group \( G \) is a nontrivial element \( \alpha \) in \( G \) such that \( g_1^{-1}\alpha g_1\dotsm g_n^{-1}\alpha g_n=1 \) for some \( g_1,\dotsm g_n\in G \); i.e., a generalized torsion element is a nontrivial element such that any product of its conjugates is trivial. A group is generalized torsion-free (GTF for short) if it doesn't contain a generalized torsion element. As an example, the fundamental group of Klein bottle \( K:=\langle a,b| abab^{-1}=1\rangle \) is torsion-free but not GTF; its element \( a \) is a generalized torsion.

A group \( G \) is bi-orderable if there is a total order \( < \) of \( G \) invariant under multiplication from both sides: \( a

In a 2015 paper, I. Chiswell, A. Glass and J. Wilson asked whether a GTF one-relator group must be bi-orderable. We prove that the group \( G=\langle a,b,d| a^{b^2}(a^b)^{-1}a^2(a^d)^{-1}a^{d^2}=1 \rangle \), where we use \( g^x \) for the conjugate \( x^{-1}gx \), is GTF and not bi-orderable. This answers their question negatively.