Left ordered groups with no non-abelian free subgroups (Q2711628)

A group \(G\) is left orderable if there is a total order \(\leq\) on \(G\) such that \(x\leq y\) implies \(zx\leq zy\) for any \(x,y,z\in G\). A group is locally indicable if every non-trivial finitely generated subgroup has an infinite cyclic quotient. Every locally indicable group is left orderable but the converse is not true as has been shown by \textit{G. M. Bergman} [Pac. J. Math. 147, 243-248 (1991; Zbl 0677.06007)] and \textit{V. M. Tararin} [Sib. Math. J. 35, 1036-1039 (1994; Zbl 0851.06006)]. However, all known examples contain a non-abelian free subgroup. In the paper, the author shows for a large class of groups not containing a non-abelian free subgroup, that any left ordered group in this class is locally indicable.