On torsion-free groups in o-minimal structures

We consider groups definable in the structure Ran and certain o-minimal expansions of it. We prove: If G = G, * is a definable abelian torsion-free group, then G is definably isomorphic to a direct sum of R, + k and R >0 , m , for some k, m 0. Futhermore, this isomorphism is definable in the structure R, +, , G . In particular, if G is semialgebraic, then the isomorphism is semialgebraic.