  Decomposable ordered structures were introduced in \cite{OnSt} to develop a
general framework to study `finite-dimensional' totally ordered structures.
This paper continues this work to include decomposable structures on which a
ordered group operation is defined on the structure. The main result at this
level of generality asserts that any such group is supersolvable, and that
topologically it is homeomorphic to the product of o-minimal groups. Then,
working in an o-minimal ordered field $\mathcal R$ satisfying some additional
assumptions, in Sections 3-7 definable ordered groups of dimension 2 and 3 are
completely analyzed modulo definable group isomorphism. Lastly, this analysis
is refined to provide a full description of these groups with respect to
definable ordered group isomorphism.
