On Borel orderable groups

We prove that any Borel Abelian ordered group B, having a countable subgroup G as the largest convex subgroup, and such that the quotient B/G is order isomorphic to R, the reals, is Borel group-order isomorphic to the product R×G, ordered lexicographically. As a main ingredient of this proof, we show, answering a question of D. Marker, that all Borel cocycles R2→Z are Borel coboundaries. A Borel classification theorem for Borel ordered ccc groups is proved.