When is a right orderable group locally indicable?

Abstract:

If a group $G$ has an ascending series $1 = G_{0} \leq G_{1} \leq \dots \leq G_{\rho } = G$ of subgroups such that for each ordinal $\alpha , G_{\alpha }\vartriangleleft G$, and $G_{\alpha + 1}/G_{\alpha }$ has no non-abelian free subsemigroup, then $G$ is right orderable if and only if it is locally indicable. In particular if $G$ is a radical-by-periodic group, then it is right orderable if and only if it is locally indicable.