On ordered groups of Morley o-rank 1

Given a cut $s$ in an ordered structure $\mathcal{M}$ we can define a localization of Morley rank—Morley o-rank, replacing each formula in definition of Morley rank with the following partial types: the cut $s$ extended with this formula. We prove in the paper that any ordered group of Morley o-rank 1 with boundedly many definable convex subgroups is weakly o-minimal and construct an example of an ordered group of Morley o-rank 1 and Morley o-degree at most 4.