Resumen:
We study the properties of the free abelian topological group A(d)(X) on a metric space (X, d) endowed with the topology generated by the Graev extension d circumflex accent of a given metric d on X. We find that the group of Lipschitz functions Lip0(X, T) is the group of continuous characters of A(d)(X). From this fact we derive some interesting properties of the metric groups A(d)(X) and Lip(0)(X, T).